Parallelogram of force

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id="toc-Newton's_proof-sublist" class="vector-toc-list"> <li id="toc-Preliminary:_the_parallelogram_of_velocity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Preliminary:_the_parallelogram_of_velocity"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Preliminary: the parallelogram of velocity</span> </div> </a> <ul id="toc-Preliminary:_the_parallelogram_of_velocity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Newton's_proof_of_the_parallelogram_of_force" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Newton's_proof_of_the_parallelogram_of_force"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Newton's proof of the parallelogram of force</span> </div> </a> <ul id="toc-Newton's_proof_of_the_parallelogram_of_force-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Bernoulli's_proof_for_perpendicular_vectors" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bernoulli's_proof_for_perpendicular_vectors"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Bernoulli's proof for perpendicular vectors</span> </div> </a> <ul id="toc-Bernoulli's_proof_for_perpendicular_vectors-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Algebraic_proof_of_the_parallelogram_of_force" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Algebraic_proof_of_the_parallelogram_of_force"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Algebraic proof of the parallelogram of force</span> </div> </a> <ul id="toc-Algebraic_proof_of_the_parallelogram_of_force-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Controversy" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" 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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Parallelogram of force</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 18 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-18" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">18 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%AA%D9%88%D8%A7%D8%B2%D9%8A_%D8%A3%D8%B6%D9%84%D8%A7%D8%B9_%D8%A7%D9%84%D9%82%D9%88%D9%89" title="متوازي أضلاع القوى – Arabic" lang="ar" hreflang="ar" data-title="متوازي أضلاع القوى" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%92%C4%83%D0%B9%D1%81%D0%B5%D0%BD_%D0%BF%D0%B0%D1%80%D0%B0%D0%BB%D0%BB%D0%B5%D0%BB%D0%BE%D0%B3%D1%80%D0%B0%D0%BC%C4%95" title="Вăйсен параллелограмĕ – Chuvash" lang="cv" hreflang="cv" data-title="Вăйсен параллелограмĕ" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Kr%C3%A6fternes_parallelogram" title="Kræfternes parallelogram – Danish" lang="da" hreflang="da" data-title="Kræfternes parallelogram" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Kr%C3%A4fteparallelogramm" title="Kräfteparallelogramm – German" lang="de" hreflang="de" data-title="Kräfteparallelogramm" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Paralelogramo_de_fuerzas" title="Paralelogramo de fuerzas – Spanish" lang="es" hreflang="es" data-title="Paralelogramo de fuerzas" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/M%C3%A9thode_du_parall%C3%A9logramme" title="Méthode du parallélogramme – French" lang="fr" hreflang="fr" data-title="Méthode du parallélogramme" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%88%D6%82%D5%AA%D5%A5%D6%80%D5%AB_%D5%A6%D5%B8%D6%82%D5%A3%D5%A1%D5%B0%D5%A5%D5%BC%D5%A1%D5%A3%D5%AB%D5%AE" title="Ուժերի զուգահեռագիծ – Armenian" lang="hy" hreflang="hy" data-title="Ուժերի զուգահեռագիծ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Paralelogram_vektora" title="Paralelogram vektora – Croatian" lang="hr" hreflang="hr" data-title="Paralelogram vektora" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A7%D7%91%D7%99%D7%9C%D7%99%D7%AA_%D7%94%D7%9B%D7%95%D7%97%D7%95%D7%AA" title="מקבילית הכוחות – Hebrew" lang="he" hreflang="he" data-title="מקבילית הכוחות" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Parallelogrammum_virium" title="Parallelogrammum virium – Latin" lang="la" hreflang="la" data-title="Parallelogrammum virium" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%8A%9B%E3%81%AE%E5%B9%B3%E8%A1%8C%E5%9B%9B%E8%BE%BA%E5%BD%A2" title="力の平行四辺形 – Japanese" lang="ja" hreflang="ja" data-title="力の平行四辺形" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Kraftparallellogram" title="Kraftparallellogram – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Kraftparallellogram" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%B0%D1%80%D0%B0%D0%BB%D0%BB%D0%B5%D0%BB%D0%BE%D0%B3%D1%80%D0%B0%D0%BC%D0%BC_%D1%81%D0%B8%D0%BB" title="Параллелограмм сил – Russian" lang="ru" hreflang="ru" data-title="Параллелограмм сил" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Paralelogram_sil" title="Paralelogram sil – Slovenian" lang="sl" hreflang="sl" data-title="Paralelogram sil" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A1%D0%BB%D0%B0%D0%B3%D0%B0%D1%9A%D0%B5_%D1%81%D0%B8%D0%BB%D0%B0" title="Слагање сила – Serbian" lang="sr" hreflang="sr" data-title="Слагање сила" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Slaganje_sila" title="Slaganje sila – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Slaganje sila" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%B5%E0%AE%BF%E0%AE%9A%E0%AF%88%E0%AE%95%E0%AE%B3%E0%AE%BF%E0%AE%A9%E0%AF%8D_%E0%AE%87%E0%AE%A3%E0%AF%88%E0%AE%95%E0%AE%B0_%E0%AE%B5%E0%AE%BF%E0%AE%A4%E0%AE%BF" title="விசைகளின் இணைகர விதி – Tamil" lang="ta" hreflang="ta" data-title="விசைகளின் இணைகர விதி" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-uk badge-Q70893996 mw-list-item" title=""><a href="https://uk.wikipedia.org/wiki/%D0%9F%D0%B0%D1%80%D0%B0%D0%BB%D0%B5%D0%BB%D0%BE%D0%B3%D1%80%D0%B0%D0%BC_%D1%81%D0%B8%D0%BB" title="Паралелограм сил – Ukrainian" lang="uk" hreflang="uk" data-title="Паралелограм сил" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a 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class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Method in physics</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Vector_parallelogram.PNG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Vector_parallelogram.PNG/200px-Vector_parallelogram.PNG" decoding="async" width="200" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Vector_parallelogram.PNG/300px-Vector_parallelogram.PNG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Vector_parallelogram.PNG/400px-Vector_parallelogram.PNG 2x" data-file-width="515" data-file-height="412" /></a><figcaption>Figure 1: Parallelogram construction for adding vectors. This construction has the same result as moving <b>F</b><sub>2</sub> so its tail coincides with the head of <b>F</b><sub>1</sub>, and taking the net force as the vector joining the tail of <b>F</b><sub>1</sub> to the head of <b>F</b><sub>2</sub>. This procedure can be repeated to add <b>F</b><sub>3</sub> to the resultant <b>F</b><sub>1</sub> + <b>F</b><sub>2</sub>, and so forth.</figcaption></figure> <p>The <b>parallelogram of forces</b> is a method for solving (or visualizing) the results of applying two <a href="/wiki/Force" title="Force">forces</a> to an object. When more than two forces are involved, the geometry is no longer a <a href="/wiki/Parallelogram" title="Parallelogram">parallelogram</a>, but the same principles apply to a <i><a href="/wiki/Polygon_of_forces" class="mw-redirect" title="Polygon of forces">polygon of forces</a></i>. The <a href="/wiki/Resultant_force" title="Resultant force">resultant force</a> due to the application of a number of forces can be found geometrically by drawing arrows for each force. The parallelogram of forces is a graphical manifestation of the <a href="/wiki/Vector_addition" class="mw-redirect" title="Vector addition">addition</a> of <a href="/wiki/Vector_(geometric)" class="mw-redirect" title="Vector (geometric)">vectors</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Newton's_proof"><span id="Newton.27s_proof"></span>Newton's proof</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parallelogram_of_force&action=edit&section=1" title="Edit section: Newton's proof"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Parallelogram_CDA_eq_BAC.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Parallelogram_CDA_eq_BAC.svg/220px-Parallelogram_CDA_eq_BAC.svg.png" decoding="async" width="220" height="110" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Parallelogram_CDA_eq_BAC.svg/330px-Parallelogram_CDA_eq_BAC.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Parallelogram_CDA_eq_BAC.svg/440px-Parallelogram_CDA_eq_BAC.svg.png 2x" data-file-width="250" data-file-height="125" /></a><figcaption>Figure 2: Parallelogram of velocity</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Preliminary:_the_parallelogram_of_velocity">Preliminary: the parallelogram of velocity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parallelogram_of_force&action=edit&section=2" title="Edit section: Preliminary: the parallelogram of velocity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Suppose a <a href="/wiki/Particle" title="Particle">particle</a> moves at a uniform rate along a line from A to B (Figure 2) in a given time (say, one <a href="/wiki/Second" title="Second">second</a>), while in the same time, the line AB moves uniformly from its position at AB to a position at DC, remaining parallel to its original orientation throughout. Accounting for both motions, the particle traces the line AC. Because a displacement in a given time is a measure of <a href="/wiki/Velocity" title="Velocity">velocity</a>, the length of AB is a measure of the particle's velocity along AB, the length of AD is a measure of the line's velocity along AD, and the length of AC is a measure of the particle's velocity along AC. The particle's motion is the same as if it had moved with a single velocity along AC.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Newton's_proof_of_the_parallelogram_of_force"><span id="Newton.27s_proof_of_the_parallelogram_of_force"></span>Newton's proof of the parallelogram of force</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parallelogram_of_force&action=edit&section=3" title="Edit section: Newton's proof of the parallelogram of force"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Suppose two <a href="/wiki/Force" title="Force">forces</a> act on a <a href="/wiki/Particle" title="Particle">particle</a> at the origin (the "tails" of the <a href="/wiki/Euclidean_vector" title="Euclidean vector">vectors</a>) of Figure 1. Let the lengths of the vectors <b>F</b><sub>1</sub> and <b>F</b><sub>2</sub> represent the <a href="/wiki/Velocity" title="Velocity">velocities</a> the two forces could produce in the particle by acting for a given time, and let the direction of each represent the direction in which they act. Each force acts independently and will produce its particular velocity whether the other force acts or not. At the end of the given time, the particle has <i>both</i> velocities. By the above proof, they are equivalent to a single velocity, <b>F</b><sub>net</sub>. By <a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's second law</a>, this vector is also a measure of the force which would produce that velocity, thus the two forces are equivalent to a single force.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Parallelogram_of_forces_-_ball_on_slope.pdf" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Parallelogram_of_forces_-_ball_on_slope.pdf/page1-220px-Parallelogram_of_forces_-_ball_on_slope.pdf.jpg" decoding="async" width="220" height="102" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Parallelogram_of_forces_-_ball_on_slope.pdf/page1-330px-Parallelogram_of_forces_-_ball_on_slope.pdf.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/64/Parallelogram_of_forces_-_ball_on_slope.pdf/page1-440px-Parallelogram_of_forces_-_ball_on_slope.pdf.jpg 2x" data-file-width="1754" data-file-height="814" /></a><figcaption>Using a parallelogram to add the forces acting on a particle on a smooth slope. We find, as we'd expect, that the resultant (double headed arrow) force acts down the slope, which will cause the particle to accelerate in that direction.</figcaption></figure> <div class="mw-heading mw-heading2"><h2 id="Bernoulli's_proof_for_perpendicular_vectors"><span id="Bernoulli.27s_proof_for_perpendicular_vectors"></span>Bernoulli's proof for perpendicular vectors</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parallelogram_of_force&action=edit&section=4" title="Edit section: Bernoulli's proof for perpendicular vectors"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>We model forces as Euclidean vectors or members of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"></span>. Our first assumption is that the resultant of two forces is in fact another force, so that for any two forces <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} ,\mathbf {G} \in \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">G</mi> </mrow> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} ,\mathbf {G} \in \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2998d6e828f66be63c9c62905bf9d7c38ebb3e83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.39ex; height:3.009ex;" alt="{\displaystyle \mathbf {F} ,\mathbf {G} \in \mathbb {R} ^{2}}"></span> there is another force <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} \oplus \mathbf {G} \in \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>⊕</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">G</mi> </mrow> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} \oplus \mathbf {G} \in \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/930cc0dc2944e2579dbd8f0ee2e11fdbe5e3ce9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.197ex; height:2.843ex;" alt="{\displaystyle \mathbf {F} \oplus \mathbf {G} \in \mathbb {R} ^{2}}"></span>. Our final assumption is that the resultant of two forces doesn't change when rotated. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R:\mathbb {R} ^{2}\to \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R:\mathbb {R} ^{2}\to \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b2cce166ae673fe421496e128ef798fb7635dd7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.78ex; height:2.676ex;" alt="{\displaystyle R:\mathbb {R} ^{2}\to \mathbb {R} ^{2}}"></span> is any rotation (any orthogonal map for the usual vector space structure of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det R=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mi>R</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det R=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/beafe52a025e83bda928ad4ce35e4f33ccc0abfd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.642ex; height:2.176ex;" alt="{\displaystyle \det R=1}"></span>), then for all forces <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} ,\mathbf {G} \in \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">G</mi> </mrow> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} ,\mathbf {G} \in \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2998d6e828f66be63c9c62905bf9d7c38ebb3e83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.39ex; height:3.009ex;" alt="{\displaystyle \mathbf {F} ,\mathbf {G} \in \mathbb {R} ^{2}}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\left(\mathbf {F} \oplus \mathbf {G} \right)=R\left(\mathbf {F} \right)\oplus R\left(\mathbf {G} \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>⊕</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">G</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>R</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>)</mo> </mrow> <mo>⊕</mo> <mi>R</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">G</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\left(\mathbf {F} \oplus \mathbf {G} \right)=R\left(\mathbf {F} \right)\oplus R\left(\mathbf {G} \right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1594190af94fab64f0a0db3f91a7a184bcb55c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.227ex; height:2.843ex;" alt="{\displaystyle R\left(\mathbf {F} \oplus \mathbf {G} \right)=R\left(\mathbf {F} \right)\oplus R\left(\mathbf {G} \right)}"></span> </p><p>Consider two perpendicular forces <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f52767d064aca73970f5a7bfc1946b2cfd77edd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.737ex; height:2.509ex;" alt="{\displaystyle \mathbf {F} _{1}}"></span> of length <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6389133972f39153163f091bc9922f5ce7385098" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.737ex; height:2.509ex;" alt="{\displaystyle \mathbf {F} _{2}}"></span> of length <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>, with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> being the length of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} _{1}\oplus \mathbf {F} _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊕</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{1}\oplus \mathbf {F} _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d692b57284e554557bbdf64898193d4ec3fc0a40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.314ex; height:2.509ex;" alt="{\displaystyle \mathbf {F} _{1}\oplus \mathbf {F} _{2}}"></span>. Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {G} _{1}:={\tfrac {a^{2}}{x^{2}}}\left(\mathbf {F} _{1}\oplus \mathbf {F} _{2}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">G</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊕</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {G} _{1}:={\tfrac {a^{2}}{x^{2}}}\left(\mathbf {F} _{1}\oplus \mathbf {F} _{2}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47f6a2016c98f77071991e12f76a4cf6b60b0f69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:20.019ex; height:4.343ex;" alt="{\displaystyle \mathbf {G} _{1}:={\tfrac {a^{2}}{x^{2}}}\left(\mathbf {F} _{1}\oplus \mathbf {F} _{2}\right)}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {G} _{2}:={\tfrac {a}{x}}R(\mathbf {F} _{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">G</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>a</mi> <mi>x</mi> </mfrac> </mstyle> </mrow> <mi>R</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {G} _{2}:={\tfrac {a}{x}}R(\mathbf {F} _{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2740078fac342181c67e96729b15e93c6b124d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.987ex; height:3.009ex;" alt="{\displaystyle \mathbf {G} _{2}:={\tfrac {a}{x}}R(\mathbf {F} _{2})}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> is the rotation between <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f52767d064aca73970f5a7bfc1946b2cfd77edd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.737ex; height:2.509ex;" alt="{\displaystyle \mathbf {F} _{1}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} _{1}\oplus \mathbf {F} _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊕</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{1}\oplus \mathbf {F} _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d692b57284e554557bbdf64898193d4ec3fc0a40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.314ex; height:2.509ex;" alt="{\displaystyle \mathbf {F} _{1}\oplus \mathbf {F} _{2}}"></span>, so <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {G_{1}} ={\tfrac {a}{x}}R\left(\mathbf {F} _{1}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>a</mi> <mi>x</mi> </mfrac> </mstyle> </mrow> <mi>R</mi> <mrow> <mo>(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {G_{1}} ={\tfrac {a}{x}}R\left(\mathbf {F} _{1}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec65260b025654a102e8967eda5dd6901c11e14e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.85ex; height:3.009ex;" alt="{\displaystyle \mathbf {G_{1}} ={\tfrac {a}{x}}R\left(\mathbf {F} _{1}\right)}"></span>. Under the invariance of the rotation, we get </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} _{1}={\frac {x}{a}}R^{-1}\left(\mathbf {G} _{1}\right)={\frac {a}{x}}R^{-1}\left(\mathbf {F} _{1}\oplus \mathbf {F} _{2}\right)={\frac {a}{x}}R^{-1}\left(\mathbf {F} _{1}\right)\oplus {\frac {a}{x}}R^{-1}\left(\mathbf {F} _{2}\right)=\mathbf {G} _{1}\oplus \mathbf {G} _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>a</mi> </mfrac> </mrow> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">G</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>x</mi> </mfrac> </mrow> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊕</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>x</mi> </mfrac> </mrow> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>⊕</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>x</mi> </mfrac> </mrow> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">G</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊕</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">G</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{1}={\frac {x}{a}}R^{-1}\left(\mathbf {G} _{1}\right)={\frac {a}{x}}R^{-1}\left(\mathbf {F} _{1}\oplus \mathbf {F} _{2}\right)={\frac {a}{x}}R^{-1}\left(\mathbf {F} _{1}\right)\oplus {\frac {a}{x}}R^{-1}\left(\mathbf {F} _{2}\right)=\mathbf {G} _{1}\oplus \mathbf {G} _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8bd912d0900ccb235140191acf9b9fb3ed2141e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:77.901ex; height:4.676ex;" alt="{\displaystyle \mathbf {F} _{1}={\frac {x}{a}}R^{-1}\left(\mathbf {G} _{1}\right)={\frac {a}{x}}R^{-1}\left(\mathbf {F} _{1}\oplus \mathbf {F} _{2}\right)={\frac {a}{x}}R^{-1}\left(\mathbf {F} _{1}\right)\oplus {\frac {a}{x}}R^{-1}\left(\mathbf {F} _{2}\right)=\mathbf {G} _{1}\oplus \mathbf {G} _{2}}"></span> </p><p>Similarly, consider two more forces <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {H} _{1}:=-\mathbf {G} _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:=</mo> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">G</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {H} _{1}:=-\mathbf {G} _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/380f98acfa0a0d2ab5ec8489a26785752495349d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.854ex; height:2.509ex;" alt="{\displaystyle \mathbf {H} _{1}:=-\mathbf {G} _{2}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {H} _{2}:={\tfrac {b^{2}}{x^{2}}}\left(\mathbf {F} _{1}\oplus \mathbf {F} _{2}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊕</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {H} _{2}:={\tfrac {b^{2}}{x^{2}}}\left(\mathbf {F} _{1}\oplus \mathbf {F} _{2}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb31bad0d0072f03bf7725c5749a4705ce90f883" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:20.01ex; height:4.343ex;" alt="{\displaystyle \mathbf {H} _{2}:={\tfrac {b^{2}}{x^{2}}}\left(\mathbf {F} _{1}\oplus \mathbf {F} _{2}\right)}"></span>. Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> be the rotation from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f52767d064aca73970f5a7bfc1946b2cfd77edd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.737ex; height:2.509ex;" alt="{\displaystyle \mathbf {F} _{1}}"></span> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {H} _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {H} _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/032dfdfbb7bed3e9df1ffe14722e06eaf46c3edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.146ex; height:2.509ex;" alt="{\displaystyle \mathbf {H} _{1}}"></span>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {H} _{1}={\tfrac {b}{x}}T\left(\mathbf {F} _{1}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>b</mi> <mi>x</mi> </mfrac> </mstyle> </mrow> <mi>T</mi> <mrow> <mo>(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {H} _{1}={\tfrac {b}{x}}T\left(\mathbf {F} _{1}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17b4d0a4731c778009b7ad549b37373a3a7da13f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.59ex; height:3.509ex;" alt="{\displaystyle \mathbf {H} _{1}={\tfrac {b}{x}}T\left(\mathbf {F} _{1}\right)}"></span>, which by inspection makes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {H} _{2}={\tfrac {b}{x}}T\left(\mathbf {F} _{2}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>b</mi> <mi>x</mi> </mfrac> </mstyle> </mrow> <mi>T</mi> <mrow> <mo>(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {H} _{2}={\tfrac {b}{x}}T\left(\mathbf {F} _{2}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d375fa5e61cc9dd227935ca6b5916639974c3c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.59ex; height:3.509ex;" alt="{\displaystyle \mathbf {H} _{2}={\tfrac {b}{x}}T\left(\mathbf {F} _{2}\right)}"></span>. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} _{2}={\frac {x}{b}}T^{-1}\left(\mathbf {H} _{2}\right)={\frac {b}{x}}T^{-1}\left(\mathbf {F} _{1}\oplus \mathbf {F} _{2}\right)={\frac {b}{x}}T^{-1}\left(\mathbf {F} _{1}\right)\oplus {\frac {b}{x}}T^{-1}\left(\mathbf {F} _{2}\right)=\mathbf {H} _{1}\oplus \mathbf {H_{2}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>b</mi> </mfrac> </mrow> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>x</mi> </mfrac> </mrow> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊕</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>x</mi> </mfrac> </mrow> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>⊕</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>x</mi> </mfrac> </mrow> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊕</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{2}={\frac {x}{b}}T^{-1}\left(\mathbf {H} _{2}\right)={\frac {b}{x}}T^{-1}\left(\mathbf {F} _{1}\oplus \mathbf {F} _{2}\right)={\frac {b}{x}}T^{-1}\left(\mathbf {F} _{1}\right)\oplus {\frac {b}{x}}T^{-1}\left(\mathbf {F} _{2}\right)=\mathbf {H} _{1}\oplus \mathbf {H_{2}} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e54840519a4ff74c74507c2145e97bfd89e768ca" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:77.82ex; height:5.509ex;" alt="{\displaystyle \mathbf {F} _{2}={\frac {x}{b}}T^{-1}\left(\mathbf {H} _{2}\right)={\frac {b}{x}}T^{-1}\left(\mathbf {F} _{1}\oplus \mathbf {F} _{2}\right)={\frac {b}{x}}T^{-1}\left(\mathbf {F} _{1}\right)\oplus {\frac {b}{x}}T^{-1}\left(\mathbf {F} _{2}\right)=\mathbf {H} _{1}\oplus \mathbf {H_{2}} }"></span> </p><p>Applying these two equations </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} _{1}\oplus \mathbf {F} _{2}=\left(\mathbf {G} _{1}\oplus \mathbf {G} _{2}\right)\oplus \left(\mathbf {H} _{1}\oplus \mathbf {H_{2}} \right)=\left(\mathbf {G} _{1}\oplus \mathbf {G} _{2}\right)\oplus \left(-\mathbf {G} _{2}\oplus \mathbf {H} _{2}\right)=\mathbf {G} _{1}\oplus \mathbf {H} _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊕</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">G</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊕</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">G</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>⊕</mo> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊕</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msub> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">G</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊕</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">G</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>⊕</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">G</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⊕</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">G</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊕</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{1}\oplus \mathbf {F} _{2}=\left(\mathbf {G} _{1}\oplus \mathbf {G} _{2}\right)\oplus \left(\mathbf {H} _{1}\oplus \mathbf {H_{2}} \right)=\left(\mathbf {G} _{1}\oplus \mathbf {G} _{2}\right)\oplus \left(-\mathbf {G} _{2}\oplus \mathbf {H} _{2}\right)=\mathbf {G} _{1}\oplus \mathbf {H} _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54267d3759b0f9b9fd1f9cb37cf346cf9af6674a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:78.174ex; height:2.843ex;" alt="{\displaystyle \mathbf {F} _{1}\oplus \mathbf {F} _{2}=\left(\mathbf {G} _{1}\oplus \mathbf {G} _{2}\right)\oplus \left(\mathbf {H} _{1}\oplus \mathbf {H_{2}} \right)=\left(\mathbf {G} _{1}\oplus \mathbf {G} _{2}\right)\oplus \left(-\mathbf {G} _{2}\oplus \mathbf {H} _{2}\right)=\mathbf {G} _{1}\oplus \mathbf {H} _{2}}"></span> </p><p>Since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {G} _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">G</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {G} _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e2417585efaa5ec47f65f0fa78b8201ac46afec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.155ex; height:2.509ex;" alt="{\displaystyle \mathbf {G} _{1}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {H} _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {H} _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09af0e33599d74dfad792813ada4b2c244d3ecd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.146ex; height:2.509ex;" alt="{\displaystyle \mathbf {H} _{2}}"></span> both lie along <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} _{1}\oplus \mathbf {F} _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊕</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{1}\oplus \mathbf {F} _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d692b57284e554557bbdf64898193d4ec3fc0a40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.314ex; height:2.509ex;" alt="{\displaystyle \mathbf {F} _{1}\oplus \mathbf {F} _{2}}"></span>, their lengths are equal <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=\left|\mathbf {F} _{1}\oplus \mathbf {F} _{2}\right|=\left|\mathbf {G} _{1}\oplus \mathbf {H} _{2}\right|={\tfrac {a^{2}}{x}}+{\tfrac {b^{2}}{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊕</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">G</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊕</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>x</mi> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>x</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=\left|\mathbf {F} _{1}\oplus \mathbf {F} _{2}\right|=\left|\mathbf {G} _{1}\oplus \mathbf {H} _{2}\right|={\tfrac {a^{2}}{x}}+{\tfrac {b^{2}}{x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5db65a24ca8a0a312bf008104a7bcb617cbda1f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:38.419ex; height:3.843ex;" alt="{\displaystyle x=\left|\mathbf {F} _{1}\oplus \mathbf {F} _{2}\right|=\left|\mathbf {G} _{1}\oplus \mathbf {H} _{2}\right|={\tfrac {a^{2}}{x}}+{\tfrac {b^{2}}{x}}}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x={\sqrt {a^{2}+b^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\sqrt {a^{2}+b^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72efb8ce57e1dfed401801c362d297a72df0f04f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.928ex; height:3.509ex;" alt="{\displaystyle x={\sqrt {a^{2}+b^{2}}}}"></span> </p><p>which implies that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} _{1}\oplus \mathbf {F} _{2}=a\mathbf {e} _{1}\oplus b\mathbf {e} _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊕</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mi>a</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊕</mo> <mi>b</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{1}\oplus \mathbf {F} _{2}=a\mathbf {e} _{1}\oplus b\mathbf {e} _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65f6dcd6cb0bcbadc694a9af1adaac863f29bfce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.039ex; height:2.509ex;" alt="{\displaystyle \mathbf {F} _{1}\oplus \mathbf {F} _{2}=a\mathbf {e} _{1}\oplus b\mathbf {e} _{2}}"></span> has length <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {a^{2}+b^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {a^{2}+b^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/460372bc2a2886a1a99b9280394eb32ec5c4fea4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.5ex; height:3.509ex;" alt="{\displaystyle {\sqrt {a^{2}+b^{2}}}}"></span>, which is the length of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\mathbf {e} _{1}+b\mathbf {e} _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>b</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\mathbf {e} _{1}+b\mathbf {e} _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20db4d8cf2ee6c43c59fe161a212f1dab6be3163" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.627ex; height:2.509ex;" alt="{\displaystyle a\mathbf {e} _{1}+b\mathbf {e} _{2}}"></span>. Thus for the case where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f52767d064aca73970f5a7bfc1946b2cfd77edd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.737ex; height:2.509ex;" alt="{\displaystyle \mathbf {F} _{1}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6389133972f39153163f091bc9922f5ce7385098" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.737ex; height:2.509ex;" alt="{\displaystyle \mathbf {F} _{2}}"></span> are perpendicular, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} _{1}\oplus \mathbf {F} _{2}=\mathbf {F} _{1}+\mathbf {F} _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊕</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{1}\oplus \mathbf {F} _{2}=\mathbf {F} _{1}+\mathbf {F} _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e75ad4bae6c65a1c71e3e7b70a879d0ffaf92e3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.727ex; height:2.509ex;" alt="{\displaystyle \mathbf {F} _{1}\oplus \mathbf {F} _{2}=\mathbf {F} _{1}+\mathbf {F} _{2}}"></span>. However, when combining our two sets of auxiliary forces we used the associativity of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oplus }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊕</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oplus }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b16e2bdaefee9eed86d866e6eba3ac47c710f60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \oplus }"></span>. Using this additional assumption, we will form an additional proof below.<sup id="cite_ref-Spivak_3-0" class="reference"><a href="#cite_note-Spivak-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> <sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Algebraic_proof_of_the_parallelogram_of_force">Algebraic proof of the parallelogram of force</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parallelogram_of_force&action=edit&section=5" title="Edit section: Algebraic proof of the parallelogram of force"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>We model forces as Euclidean vectors or members of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"></span>. Our first assumption is that the resultant of two forces is in fact another force, so that for any two forces <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} ,\mathbf {G} \in \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">G</mi> </mrow> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} ,\mathbf {G} \in \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2998d6e828f66be63c9c62905bf9d7c38ebb3e83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.39ex; height:3.009ex;" alt="{\displaystyle \mathbf {F} ,\mathbf {G} \in \mathbb {R} ^{2}}"></span> there is another force <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} \oplus \mathbf {G} \in \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>⊕</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">G</mi> </mrow> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} \oplus \mathbf {G} \in \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/930cc0dc2944e2579dbd8f0ee2e11fdbe5e3ce9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.197ex; height:2.843ex;" alt="{\displaystyle \mathbf {F} \oplus \mathbf {G} \in \mathbb {R} ^{2}}"></span>. We assume commutativity, as these are forces being applied concurrently, so the order shouldn't matter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} \oplus \mathbf {G} =\mathbf {G} \oplus \mathbf {F} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>⊕</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">G</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">G</mi> </mrow> <mo>⊕</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} \oplus \mathbf {G} =\mathbf {G} \oplus \mathbf {F} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74049d3de8a88b4e8635e27d3874e1ddb1203a17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.346ex; height:2.343ex;" alt="{\displaystyle \mathbf {F} \oplus \mathbf {G} =\mathbf {G} \oplus \mathbf {F} }"></span>. </p><p>Consider the map <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b)=a\mathbf {e} _{1}+b\mathbf {e} _{2}\mapsto a\mathbf {e} _{1}\oplus b\mathbf {e} _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>b</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">↦</mo> <mi>a</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊕</mo> <mi>b</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b)=a\mathbf {e} _{1}+b\mathbf {e} _{2}\mapsto a\mathbf {e} _{1}\oplus b\mathbf {e} _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5469989a36326ef1b9e7b4f2df07ed732758cf8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.036ex; height:2.843ex;" alt="{\displaystyle (a,b)=a\mathbf {e} _{1}+b\mathbf {e} _{2}\mapsto a\mathbf {e} _{1}\oplus b\mathbf {e} _{2}}"></span> </p><p>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oplus }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊕</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oplus }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b16e2bdaefee9eed86d866e6eba3ac47c710f60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \oplus }"></span> is associative, then this map will be linear. Since it also sends <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {e} _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {e} _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4aed8f473b59ea2b05d5720248b65333c35d3f0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.279ex; height:2.009ex;" alt="{\displaystyle \mathbf {e} _{1}}"></span> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {e} _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {e} _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4aed8f473b59ea2b05d5720248b65333c35d3f0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.279ex; height:2.009ex;" alt="{\displaystyle \mathbf {e} _{1}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {e} _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {e} _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5afa38f879249b10909c9dad8e9acf54dfa76286" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.279ex; height:2.009ex;" alt="{\displaystyle \mathbf {e} _{2}}"></span> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {e} _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {e} _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5afa38f879249b10909c9dad8e9acf54dfa76286" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.279ex; height:2.009ex;" alt="{\displaystyle \mathbf {e} _{2}}"></span>, it must also be the identity map. Thus <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oplus }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊕</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oplus }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b16e2bdaefee9eed86d866e6eba3ac47c710f60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \oplus }"></span> must be equivalent to the normal vector addition operator.<sup id="cite_ref-Spivak_3-1" class="reference"><a href="#cite_note-Spivak-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Controversy">Controversy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parallelogram_of_force&action=edit&section=6" title="Edit section: Controversy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Disputed plainlinks metadata ambox ambox-content ambox-disputed" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/System-search.svg/45px-System-search.svg.png" decoding="async" width="45" height="45" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/System-search.svg/68px-System-search.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/17/System-search.svg/90px-System-search.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section's <b>factual accuracy is <a href="/wiki/Wikipedia:Accuracy_dispute" title="Wikipedia:Accuracy dispute">disputed</a></b>.<span class="hide-when-compact"> Relevant discussion may be found on the <a href="/wiki/Talk:Parallelogram_of_force#Controversy_Section" title="Talk:Parallelogram of force">talk page</a>. Please help to ensure that disputed statements are <a href="/wiki/Wikipedia:Reliable_sources" title="Wikipedia:Reliable sources">reliably sourced</a>.</span> <span class="date-container"><i>(<span class="date">September 2024</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>The mathematical proof of the parallelogram of force is not generally accepted to be mathematically valid. Various proofs were developed (chiefly <i>Duchayla's</i> and <i><a href="/wiki/Sim%C3%A9on_Denis_Poisson" title="Siméon Denis Poisson">Poisson's</a></i>), and these also caused objections. That the parallelogram of force was true was not questioned, but <i>why</i> it was true. Today the parallelogram of force is accepted as an empirical fact, non-reducible to Newton's first principles. <sup id="cite_ref-Spivak_3-2" class="reference"><a href="#cite_note-Spivak-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> <sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parallelogram_of_force&action=edit&section=7" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="https://en.wikisource.org/wiki/The_Mathematical_Principles_of_Natural_Philosophy_(1729)/Axioms,_or_Laws_of_Motion" class="extiw" title="s:The Mathematical Principles of Natural Philosophy (1729)/Axioms, or Laws of Motion">Newton's <i>Mathematical Principles of Natural Philosophy</i>, Axioms or Laws of Motion, Corollary I</a>, at <a href="https://en.wikisource.org/wiki/Main_Page" class="extiw" title="s:Main Page">Wikisource</a></li> <li><a href="/wiki/Vector_(geometric)" class="mw-redirect" title="Vector (geometric)">Vector (geometric)</a></li> <li><a href="/wiki/Net_force" title="Net force">Net force</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parallelogram_of_force&action=edit&section=8" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and 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Cambridge University Press. p. <a rel="nofollow" class="external text" href="https://archive.org/details/atreatiseonanal00routgoog/page/n22">6</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Treatise+on+Analytical+Statics&rft.pages=6&rft.pub=Cambridge+University+Press&rft.date=1896&rft.aulast=Routh&rft.aufirst=Edward+John&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fatreatiseonanal00routgoog&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParallelogram+of+force" class="Z3988"></span>, at <a rel="nofollow" class="external text" href="https://books.google.com/books">Google books</a></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"> Routh (1896), p. 14</span> </li> <li id="cite_note-Spivak-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-Spivak_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Spivak_3-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Spivak_3-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSpivak2010" class="citation book cs1"><a href="/wiki/Michael_Spivak" title="Michael Spivak">Spivak, Michael</a> (2010). <i>Mechanics I</i>. 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Publish or Perish, Inc. pp. 278–282. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-914098-32-4" title="Special:BookSources/978-0-914098-32-4"><bdi>978-0-914098-32-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mechanics+I&rft.series=Physics+for+Mathematicians&rft.pages=278-282&rft.pub=Publish+or+Perish%2C+Inc.&rft.date=2010&rft.isbn=978-0-914098-32-4&rft.aulast=Spivak&rft.aufirst=Michael&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParallelogram+of+force" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBernoulli1728" class="citation book cs1"><a href="/wiki/Daniel_Bernoulli" title="Daniel Bernoulli">Bernoulli, Daniel</a> (1728). <i>Examen principiorum mechanicae et demonstrationes geometricae de compositione et resolutione virium</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Examen+principiorum+mechanicae+et+demonstrationes+geometricae+de+compositione+et+resolutione+virium&rft.date=1728&rft.aulast=Bernoulli&rft.aufirst=Daniel&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParallelogram+of+force" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMach1974" class="citation book cs1"><a href="/wiki/Ernest_Mach" class="mw-redirect" title="Ernest Mach">Mach, Ernest</a> (1974). <i>The Science of Mechanics</i>. Open Court Publishing Co. pp. 55–57.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Science+of+Mechanics&rft.pages=55-57&rft.pub=Open+Court+Publishing+Co.&rft.date=1974&rft.aulast=Mach&rft.aufirst=Ernest&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParallelogram+of+force" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLange2009" class="citation web cs1">Lange, Marc (2009). <a rel="nofollow" class="external text" href="https://philosophy.unc.edu/wp-content/uploads/sites/122/2013/10/tale-of-two-vectors-published.pdf">"A Tale of Two Vectors"</a> <span class="cs1-format">(PDF)</span>. <i>Dialectica, 63</i>. pp. 397–431.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Dialectica%2C+63&rft.atitle=A+Tale+of+Two+Vectors&rft.pages=397-431&rft.date=2009&rft.aulast=Lange&rft.aufirst=Marc&rft_id=https%3A%2F%2Fphilosophy.unc.edu%2Fwp-content%2Fuploads%2Fsites%2F122%2F2013%2F10%2Ftale-of-two-vectors-published.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParallelogram+of+force" class="Z3988"></span></span> </li> </ol></div></div> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl 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.mw-parser-output .navbox{display:none!important}}</style><style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style></div><div role="navigation" class="navbox" aria-labelledby="Sir_Isaac_Newton" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Isaac_Newton" title="Template:Isaac Newton"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Isaac_Newton" title="Template talk:Isaac Newton"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Isaac_Newton" title="Special:EditPage/Template:Isaac Newton"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Sir_Isaac_Newton" style="font-size:114%;margin:0 4em"><a href="/wiki/Isaac_Newton" title="Isaac Newton">Sir Isaac Newton</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;">Publications</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><i><a href="/wiki/Method_of_Fluxions" title="Method of Fluxions">Fluxions</a></i> (1671)</li> <li><i><a href="/wiki/De_motu_corporum_in_gyrum" title="De motu corporum in gyrum">De Motu</a></i> (1684)</li> <li><i><a href="/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematica" title="Philosophiæ Naturalis Principia Mathematica">Principia</a></i> (1687)</li> <li><i><a href="/wiki/Opticks" title="Opticks">Opticks</a></i> (1704)</li> <li><i><a href="/wiki/The_Queries" class="mw-redirect" title="The Queries">Queries</a></i> (1704)</li> <li><i><a href="/wiki/Arithmetica_Universalis" title="Arithmetica Universalis">Arithmetica</a></i> (1707)</li> <li><i><a href="/wiki/De_analysi_per_aequationes_numero_terminorum_infinitas" title="De analysi per aequationes numero terminorum infinitas">De Analysi</a></i> (1711)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;">Other writings</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><i><a href="/wiki/Quaestiones_quaedam_philosophicae" title="Quaestiones quaedam philosophicae">Quaestiones</a></i> (1661–1665)</li> <li>"<a href="/wiki/Standing_on_the_shoulders_of_giants" title="Standing on the shoulders of giants">standing on the shoulders of giants</a>" (1675)</li> <li><i><a href="/wiki/Notes_on_the_Jewish_Temple" title="Notes on the Jewish Temple">Notes on the Jewish Temple</a></i> (c. 1680)</li> <li>"<a href="/wiki/General_Scholium" title="General Scholium">General Scholium</a>" (1713; <i>"<a href="/wiki/Hypotheses_non_fingo" title="Hypotheses non fingo">hypotheses non fingo</a>"</i> )</li> <li><i><a href="/wiki/The_Chronology_of_Ancient_Kingdoms_Amended" title="The Chronology of Ancient Kingdoms Amended">Ancient Kingdoms Amended</a></i> (1728)</li> <li><i><a href="/wiki/An_Historical_Account_of_Two_Notable_Corruptions_of_Scripture" title="An Historical Account of Two Notable Corruptions of Scripture">Corruptions of Scripture</a></i> (1754)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;">Contributions</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Calculus" title="Calculus">Calculus</a> <ul><li><a href="/wiki/Fluxion" title="Fluxion">fluxion</a></li></ul></li> <li><a href="/wiki/Impact_depth" title="Impact depth">Impact depth</a></li> <li><a href="/wiki/Inertia" title="Inertia">Inertia</a></li> <li><a href="/wiki/Newton_disc" title="Newton disc">Newton disc</a></li> <li><a href="/wiki/Newton_polygon" title="Newton polygon">Newton polygon</a> <ul><li><a href="/wiki/Newton%E2%80%93Okounkov_body" title="Newton–Okounkov body">Newton–Okounkov body</a></li></ul></li> <li><a href="/wiki/Newton%27s_reflector" title="Newton's reflector">Newton's reflector</a></li> <li><a href="/wiki/Newtonian_telescope" title="Newtonian telescope">Newtonian telescope</a></li> <li><a href="/wiki/Newton_scale" title="Newton scale">Newton scale</a></li> <li><a href="/wiki/Newton%27s_metal" title="Newton's metal">Newton's metal</a></li> <li><a href="/wiki/Spectrum" title="Spectrum">Spectrum</a></li> <li><a href="/wiki/Structural_coloration" title="Structural coloration">Structural coloration</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;"><a href="/wiki/Newtonianism" title="Newtonianism">Newtonianism</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bucket_argument" title="Bucket argument">Bucket argument</a></li> <li><a href="/wiki/Newton%27s_inequalities" title="Newton's inequalities">Newton's inequalities</a></li> <li><a href="/wiki/Newton%27s_law_of_cooling" title="Newton's law of cooling">Newton's law of cooling</a></li> <li><a href="/wiki/Newton%27s_law_of_universal_gravitation" title="Newton's law of universal gravitation">Newton's law of universal gravitation</a> <ul><li><a href="/wiki/Post-Newtonian_expansion" title="Post-Newtonian expansion">post-Newtonian expansion</a></li> <li><a href="/wiki/Parameterized_post-Newtonian_formalism" title="Parameterized post-Newtonian formalism">parameterized</a></li> <li><a href="/wiki/Gravitational_constant" title="Gravitational constant">gravitational constant</a></li></ul></li> <li><a href="/wiki/Newton%E2%80%93Cartan_theory" title="Newton–Cartan theory">Newton–Cartan theory</a></li> <li><a href="/wiki/Schr%C3%B6dinger%E2%80%93Newton_equation" title="Schrödinger–Newton equation">Schrödinger–Newton equation</a></li> <li><a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a> <ul><li><a href="/wiki/Kepler%27s_laws_of_planetary_motion" title="Kepler's laws of planetary motion">Kepler's laws</a></li></ul></li> <li><a href="/wiki/Newtonian_dynamics" title="Newtonian dynamics">Newtonian dynamics</a></li> <li><a href="/wiki/Newton%27s_method_in_optimization" title="Newton's method in optimization">Newton's method in optimization</a> <ul><li><a href="/wiki/Problem_of_Apollonius" title="Problem of Apollonius">Apollonius's problem</a></li> <li><a href="/wiki/Truncated_Newton_method" title="Truncated Newton method">truncated Newton method</a></li></ul></li> <li><a href="/wiki/Gauss%E2%80%93Newton_algorithm" title="Gauss–Newton algorithm">Gauss–Newton algorithm</a></li> <li><a href="/wiki/Newton%27s_rings" title="Newton's rings">Newton's rings</a></li> <li><a href="/wiki/Newton%27s_theorem_about_ovals" title="Newton's theorem about ovals">Newton's theorem about ovals</a></li> <li><a href="/wiki/Newton%E2%80%93Pepys_problem" title="Newton–Pepys problem">Newton–Pepys problem</a></li> <li><a href="/wiki/Newtonian_potential" title="Newtonian potential">Newtonian potential</a></li> <li><a href="/wiki/Newtonian_fluid" title="Newtonian fluid">Newtonian fluid</a></li> <li><a href="/wiki/Classical_mechanics" title="Classical mechanics">Classical mechanics</a></li> <li><a href="/wiki/Corpuscular_theory_of_light" title="Corpuscular theory of light">Corpuscular theory of light</a></li> <li><a href="/wiki/Leibniz%E2%80%93Newton_calculus_controversy" title="Leibniz–Newton calculus controversy">Leibniz–Newton calculus controversy</a></li> <li><a href="/wiki/Newton%27s_notation" class="mw-redirect" title="Newton's notation">Newton's notation</a></li> <li><a href="/wiki/Rotating_spheres" title="Rotating spheres">Rotating spheres</a></li> <li><a href="/wiki/Newton%27s_cannonball" title="Newton's cannonball">Newton's cannonball</a></li> <li><a href="/wiki/Newton%E2%80%93Cotes_formulas" title="Newton–Cotes formulas">Newton–Cotes formulas</a></li> <li><a href="/wiki/Newton%27s_method" title="Newton's method">Newton's method</a> <ul><li><a href="/wiki/Generalized_Gauss%E2%80%93Newton_method" title="Generalized Gauss–Newton method">generalized Gauss–Newton method</a></li></ul></li> <li><a href="/wiki/Newton_fractal" title="Newton fractal">Newton fractal</a></li> <li><a href="/wiki/Newton%27s_identities" title="Newton's identities">Newton's identities</a></li> <li><a href="/wiki/Newton_polynomial" title="Newton polynomial">Newton polynomial</a></li> <li><a href="/wiki/Newton%27s_theorem_of_revolving_orbits" title="Newton's theorem of revolving orbits">Newton's theorem of revolving orbits</a></li> <li><a href="/wiki/Newton%E2%80%93Euler_equations" title="Newton–Euler equations">Newton–Euler equations</a></li> <li><a href="/wiki/Power_number" title="Power number">Newton number</a> <ul><li><a href="/wiki/Kissing_number" title="Kissing number">kissing number problem</a></li></ul></li> <li><a href="/wiki/Difference_quotient" title="Difference quotient">Newton's quotient</a></li> <li><a class="mw-selflink selflink">Parallelogram of force</a></li> <li><a href="/wiki/Puiseux_series" title="Puiseux series">Newton–Puiseux theorem</a></li> <li><a href="/wiki/Absolute_space_and_time#Newton" title="Absolute space and time">Absolute space and time</a></li> <li><a href="/wiki/Luminiferous_aether" title="Luminiferous aether">Luminiferous aether</a></li> <li><a href="/wiki/Finite_difference" title="Finite difference">Newtonian series</a> <ul><li><a href="/wiki/Table_of_Newtonian_series" title="Table of Newtonian series">table</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;">Personal life</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Woolsthorpe_Manor" title="Woolsthorpe Manor">Woolsthorpe Manor</a> (birthplace)</li> <li><a href="/wiki/Cranbury_Park" title="Cranbury Park">Cranbury Park</a> (home)</li> <li><a href="/wiki/Early_life_of_Isaac_Newton" title="Early life of Isaac Newton">Early life</a></li> <li><a href="/wiki/Later_life_of_Isaac_Newton" title="Later life of Isaac Newton">Later life</a></li> <li><a href="/wiki/Isaac_Newton%27s_apple_tree" title="Isaac Newton's apple tree">Apple tree</a></li> <li><a href="/wiki/Religious_views_of_Isaac_Newton" title="Religious views of Isaac Newton">Religious views</a></li> <li><a href="/wiki/Isaac_Newton%27s_occult_studies" title="Isaac Newton's occult studies">Occult studies</a></li> <li><a href="/wiki/Scientific_Revolution" title="Scientific Revolution">Scientific Revolution</a></li> <li><a href="/wiki/Copernican_Revolution" title="Copernican Revolution">Copernican Revolution</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;">Relations</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Catherine_Barton" title="Catherine Barton">Catherine Barton</a> (niece)</li> <li><a href="/wiki/John_Conduitt" title="John Conduitt">John Conduitt</a> (nephew-in-law)</li> <li><a href="/wiki/Isaac_Barrow" title="Isaac Barrow">Isaac Barrow</a> (professor)</li> <li><a href="/wiki/William_Clarke_(apothecary)" title="William Clarke (apothecary)">William Clarke</a> (mentor)</li> <li><a href="/wiki/Benjamin_Pulleyn" title="Benjamin Pulleyn">Benjamin Pulleyn</a> (tutor)</li> <li><a href="/wiki/Roger_Cotes" title="Roger Cotes">Roger Cotes</a> (student)</li> <li><a href="/wiki/William_Whiston" title="William Whiston">William Whiston</a> (student)</li> <li><a href="/wiki/John_Keill" title="John Keill">John Keill</a> (disciple)</li> <li><a href="/wiki/William_Stukeley" title="William Stukeley">William Stukeley</a> (friend)</li> <li><a href="/wiki/William_Jones_(mathematician)" title="William Jones (mathematician)">William Jones</a> (friend)</li> <li><a href="/wiki/Abraham_de_Moivre" title="Abraham de Moivre">Abraham de Moivre</a> (friend)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;"><a href="/wiki/Isaac_Newton_in_popular_culture" title="Isaac Newton in popular culture">Depictions</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Newton_(Blake)" title="Newton (Blake)"><i>Newton</i> by Blake</a> (monotype)</li> <li><a href="/wiki/Newton_(Paolozzi)" title="Newton (Paolozzi)"><i>Newton</i> by Paolozzi</a> (sculpture)</li> <li><i><a href="/wiki/Isaac_Newton_Gargoyle" title="Isaac Newton Gargoyle">Isaac Newton Gargoyle</a></i></li> <li><i><a href="/wiki/Astronomers_Monument" title="Astronomers Monument">Astronomers Monument</a></i></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;"><a href="/wiki/List_of_things_named_after_Isaac_Newton" title="List of things named after Isaac Newton">Namesake</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Newton_(unit)" title="Newton (unit)">Newton (unit)</a></li> <li><a href="/wiki/Newton%27s_cradle" title="Newton's cradle">Newton's cradle</a></li> <li><a href="/wiki/Isaac_Newton_Institute" title="Isaac Newton Institute">Isaac Newton Institute</a></li> <li><a href="/wiki/Institute_of_Physics_Isaac_Newton_Medal" class="mw-redirect" title="Institute of Physics Isaac Newton Medal">Isaac Newton Medal</a></li> <li><a href="/wiki/Isaac_Newton_Telescope" title="Isaac Newton Telescope">Isaac Newton Telescope</a></li> <li><a href="/wiki/Isaac_Newton_Group_of_Telescopes" title="Isaac Newton Group of Telescopes">Isaac Newton Group of Telescopes</a></li> <li><a href="/wiki/XMM-Newton" title="XMM-Newton">XMM-Newton</a></li> <li><a href="/wiki/Sir_Isaac_Newton_Sixth_Form" title="Sir Isaac Newton Sixth Form">Sir Isaac Newton Sixth Form</a></li> <li><a href="/wiki/Statal_Institute_of_Higher_Education_Isaac_Newton" title="Statal Institute of Higher Education Isaac Newton">Statal Institute of Higher Education Isaac Newton</a></li> <li><a href="/wiki/Newton_International_Fellowship" title="Newton International Fellowship">Newton International Fellowship</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;">Categories</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"><div class="div-col"> <div class="CategoryTreeTag" data-ct-options="{"mode":20,"hideprefix":20,"showcount":false,"namespaces":false,"notranslations":false}"><div class="CategoryTreeSection"><div class="CategoryTreeItem"><span class="CategoryTreeBullet"><a class="CategoryTreeToggle" data-ct-title="Isaac_Newton" aria-expanded="false"></a> </span> <bdi dir="ltr"><a href="/wiki/Category:Isaac_Newton" title="Category:Isaac 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Parallelogram of force - Wikipedia