Direction cosine - Wikipedia

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Available in 16 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-16" 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">16 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%A6%E0%A6%BF%E0%A6%95_%E0%A6%95%E0%A7%8B%E0%A6%B8%E0%A6%BE%E0%A6%87%E0%A6%A8" title="দিক কোসাইন – Bangla" lang="bn" hreflang="bn" data-title="দিক কোসাইন" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Cosinus_director" title="Cosinus director – Catalan" lang="ca" hreflang="ca" data-title="Cosinus director" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Richtungskosinus" title="Richtungskosinus – German" lang="de" hreflang="de" data-title="Richtungskosinus" 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/Cosenos_directores" title="Cosenos directores – Spanish" lang="es" hreflang="es" data-title="Cosenos directores" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%DA%A9%D8%B3%DB%8C%D9%86%D9%88%D8%B3_%D9%87%D8%A7%D8%AF%DB%8C" title="کسینوس هادی – Persian" lang="fa" hreflang="fa" data-title="کسینوس هادی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Cosinus_directeur" title="Cosinus directeur – French" lang="fr" hreflang="fr" data-title="Cosinus directeur" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Coseni_direttori" title="Coseni direttori – Italian" lang="it" hreflang="it" data-title="Coseni direttori" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%91%D0%B0%D2%93%D1%8B%D1%82%D1%82%D0%B0%D1%83%D1%88%D1%8B_%D0%BA%D0%BE%D1%81%D0%B8%D0%BD%D1%83%D1%81%D1%82%D0%B0%D1%80" title="Бағыттаушы косинустар – Kazakh" lang="kk" hreflang="kk" data-title="Бағыттаушы косинустар" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Retningscosinus" title="Retningscosinus – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Retningscosinus" 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-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Cosinusy_kierunkowe" title="Cosinusy kierunkowe – Polish" lang="pl" hreflang="pl" data-title="Cosinusy kierunkowe" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Cosinus_director" title="Cosinus director – Romanian" lang="ro" hreflang="ro" data-title="Cosinus director" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Riktningscosiner" title="Riktningscosiner – Swedish" lang="sv" hreflang="sv" data-title="Riktningscosiner" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A4%E0%AE%BF%E0%AE%9A%E0%AF%88%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AF%8A%E0%AE%9A%E0%AF%88%E0%AE%A9%E0%AF%8D%E0%AE%95%E0%AE%B3%E0%AF%8D" title="திசைக்கொசைன்கள் – Tamil" lang="ta" hreflang="ta" data-title="திசைக்கொசைன்கள்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9D%D0%B0%D0%BF%D1%80%D1%8F%D0%BC%D0%BD%D1%96_%D0%BA%D0%BE%D1%81%D0%B8%D0%BD%D1%83%D1%81%D0%B8" title="Напрямні косинуси – Ukrainian" lang="uk" hreflang="uk" data-title="Напрямні косинуси" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Cosin_ch%E1%BB%89_h%C6%B0%E1%BB%9Bng" title="Cosin chỉ hướng – Vietnamese" lang="vi" hreflang="vi" data-title="Cosin chỉ hướng" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%96%B9%E5%90%91%E9%A4%98%E5%BC%A6" title="方向餘弦 – Chinese" lang="zh" hreflang="zh" data-title="方向餘弦" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q2518906#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu 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data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" 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">Cosines of the angles between a vector and the coordinate axes</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-No_footnotes plainlinks metadata ambox ambox-style ambox-No_footnotes" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/40px-Text_document_with_red_question_mark.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/60px-Text_document_with_red_question_mark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/80px-Text_document_with_red_question_mark.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 article includes a <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">list of references</a>, <a href="/wiki/Wikipedia:Further_reading" title="Wikipedia:Further reading">related reading</a>, or <a href="/wiki/Wikipedia:External_links" title="Wikipedia:External links">external links</a>, <b>but its sources remain unclear because it lacks <a href="/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">inline citations</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Wikipedia:WikiProject_Fact_and_Reference_Check" class="mw-redirect" title="Wikipedia:WikiProject Fact and Reference Check">improve</a> this article by <a href="/wiki/Wikipedia:When_to_cite" title="Wikipedia:When to cite">introducing</a> more precise citations.</span> <span class="date-container"><i>(<span class="date">January 2017</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>In <a href="/wiki/Analytic_geometry" title="Analytic geometry">analytic geometry</a>, the <b>direction cosines</b> (or <b>directional cosines</b>) of a <a href="/wiki/Euclidean_vector" title="Euclidean vector">vector</a> are the <a href="/wiki/Cosine" class="mw-redirect" title="Cosine">cosines</a> of the angles between the vector and the three positive coordinate axes. Equivalently, they are the contributions of each component of the <a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a> to a <a href="/wiki/Unit_vector" title="Unit vector">unit vector</a> in that <a href="/wiki/Direction_(geometry)" title="Direction (geometry)">direction</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Three-dimensional_Cartesian_coordinates">Three-dimensional Cartesian coordinates</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Direction_cosine&action=edit&section=1" title="Edit section: Three-dimensional Cartesian coordinates"><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:Direction_cosine_vector.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Direction_cosine_vector.svg/220px-Direction_cosine_vector.svg.png" decoding="async" width="220" height="194" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Direction_cosine_vector.svg/330px-Direction_cosine_vector.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Direction_cosine_vector.svg/440px-Direction_cosine_vector.svg.png 2x" data-file-width="402" data-file-height="354" /></a><figcaption>Vector <span class="texhtml"><b>v</b></span> in <span class="nowrap">⁠<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} ^{3}}"> <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>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" 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} ^{3}}"></span>⁠</span></figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Direction_cosine_unit_vector.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/Direction_cosine_unit_vector.svg/220px-Direction_cosine_unit_vector.svg.png" decoding="async" width="220" height="199" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/Direction_cosine_unit_vector.svg/330px-Direction_cosine_unit_vector.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/Direction_cosine_unit_vector.svg/440px-Direction_cosine_unit_vector.svg.png 2x" data-file-width="391" data-file-height="354" /></a><figcaption>Direction cosines and direction angles for the unit vector <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 {\tfrac {\mathbf {v} }{|\mathbf {v} |}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\mathbf {v} }{|\mathbf {v} |}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c36429b6b1b46996c4f449ec15d801927954673f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.749ex; height:3.843ex;" alt="{\displaystyle {\tfrac {\mathbf {v} }{|\mathbf {v} |}}}"></span></figcaption></figure> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">Cartesian coordinates</a></div> <p>If <span class="texhtml"><b>v</b></span> is a <a href="/wiki/Euclidean_vector" title="Euclidean vector">Euclidean vector</a> in <a href="/wiki/Three-dimensional_space" title="Three-dimensional space">three-dimensional</a> <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>, <span class="nowrap">⁠<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} ^{3},}"> <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>3</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/deb17c1074c77de2cf88d45bcd6d7a795b0f5d44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.379ex; height:3.009ex;" alt="{\displaystyle \mathbb {R} ^{3},}"></span>⁠</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 {v} =v_{x}\mathbf {e} _{x}+v_{y}\mathbf {e} _{y}+v_{z}\mathbf {e} _{z},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} =v_{x}\mathbf {e} _{x}+v_{y}\mathbf {e} _{y}+v_{z}\mathbf {e} _{z},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ee63b95240fa17f06788a3ad928110d27eb7287" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.342ex; height:2.676ex;" alt="{\displaystyle \mathbf {v} =v_{x}\mathbf {e} _{x}+v_{y}\mathbf {e} _{y}+v_{z}\mathbf {e} _{z},}"></span> </p><p>where <span class="texhtml"><b>e</b><sub><i>x</i></sub>, <b>e</b><sub><i>y</i></sub>, <b>e</b><sub><i>z</i></sub></span> are the <a href="/wiki/Standard_basis" title="Standard basis">standard basis</a> in Cartesian notation, then the direction cosines are </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 {\begin{alignedat}{2}\alpha &{}=\cos a={\frac {\mathbf {v} \cdot \mathbf {e} _{x}}{\Vert \mathbf {v} \Vert }}&&{}={\frac {v_{x}}{\sqrt {v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}}},\\\beta &{}=\cos b={\frac {\mathbf {v} \cdot \mathbf {e} _{y}}{\Vert \mathbf {v} \Vert }}&&{}={\frac {v_{y}}{\sqrt {v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}}},\\\gamma &{}=\cos c={\frac {\mathbf {v} \cdot \mathbf {e} _{z}}{\Vert \mathbf {v} \Vert }}&&{}={\frac {v_{z}}{\sqrt {v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}}}.\end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <mi>α</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mrow> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> </mrow> </mfrac> </mrow> </mtd> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msqrt> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>β</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>b</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> <mrow> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> </mrow> </mfrac> </mrow> </mtd> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <msqrt> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>γ</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>c</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> </mrow> </mfrac> </mrow> </mtd> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <msqrt> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{2}\alpha &{}=\cos a={\frac {\mathbf {v} \cdot \mathbf {e} _{x}}{\Vert \mathbf {v} \Vert }}&&{}={\frac {v_{x}}{\sqrt {v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}}},\\\beta &{}=\cos b={\frac {\mathbf {v} \cdot \mathbf {e} _{y}}{\Vert \mathbf {v} \Vert }}&&{}={\frac {v_{y}}{\sqrt {v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}}},\\\gamma &{}=\cos c={\frac {\mathbf {v} \cdot \mathbf {e} _{z}}{\Vert \mathbf {v} \Vert }}&&{}={\frac {v_{z}}{\sqrt {v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}}}.\end{alignedat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c73274f76b023bb94737dc4df4a67afecb0d2b7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.005ex; width:38.738ex; height:23.176ex;" alt="{\displaystyle {\begin{alignedat}{2}\alpha &{}=\cos a={\frac {\mathbf {v} \cdot \mathbf {e} _{x}}{\Vert \mathbf {v} \Vert }}&&{}={\frac {v_{x}}{\sqrt {v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}}},\\\beta &{}=\cos b={\frac {\mathbf {v} \cdot \mathbf {e} _{y}}{\Vert \mathbf {v} \Vert }}&&{}={\frac {v_{y}}{\sqrt {v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}}},\\\gamma &{}=\cos c={\frac {\mathbf {v} \cdot \mathbf {e} _{z}}{\Vert \mathbf {v} \Vert }}&&{}={\frac {v_{z}}{\sqrt {v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}}}.\end{alignedat}}}"></span> </p><p>It follows that by squaring each equation and adding the results </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 \cos ^{2}a+\cos ^{2}b+\cos ^{2}c=\alpha ^{2}+\beta ^{2}+\gamma ^{2}=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>a</mi> <mo>+</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>b</mi> <mo>+</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>c</mi> <mo>=</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos ^{2}a+\cos ^{2}b+\cos ^{2}c=\alpha ^{2}+\beta ^{2}+\gamma ^{2}=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/890d231814bd1b5a20189e15005a114a7e611879" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.527ex; height:3.176ex;" alt="{\displaystyle \cos ^{2}a+\cos ^{2}b+\cos ^{2}c=\alpha ^{2}+\beta ^{2}+\gamma ^{2}=1.}"></span> </p><p>Here <span class="texhtml"><i>α</i>, <i>β</i>, <i>γ</i></span> are the direction cosines and the Cartesian coordinates of the <a href="/wiki/Unit_vector" title="Unit vector">unit vector</a> <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 {\tfrac {\mathbf {v} }{|\mathbf {v} |}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mstyle> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\mathbf {v} }{|\mathbf {v} |}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f49ace27d533f5e9eda893d8c1d59ade54fb606" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.395ex; height:3.843ex;" alt="{\displaystyle {\tfrac {\mathbf {v} }{|\mathbf {v} |}},}"></span> and <span class="texhtml"><i>a</i>, <i>b</i>, <i>c</i></span> are the direction angles of the vector <span class="texhtml"><b>v</b></span>. </p><p>The direction angles <span class="texhtml"><i>a</i>, <i>b</i>, <i>c</i></span> are <a href="/wiki/Acute_angle" class="mw-redirect" title="Acute angle">acute</a> or <a href="/wiki/Obtuse_angle" class="mw-redirect" title="Obtuse angle">obtuse angles</a>, i.e., <span class="texhtml">0 ≤ <i>a</i> ≤ π</span>, <span class="texhtml">0 ≤ <i>b</i> ≤ <i>π</i></span> and <span class="texhtml">0 ≤ <i>c</i> ≤ <i>π</i></span>, and they denote the angles formed between <span class="texhtml"><b>v</b></span> and the unit basis vectors <span class="texhtml"><b>e</b><sub><i>x</i></sub>, <b>e</b><sub><i>y</i></sub>, <b>e</b><sub><i>z</i></sub></span>. </p> <div class="mw-heading mw-heading2"><h2 id="General_meaning">General meaning</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Direction_cosine&action=edit&section=2" title="Edit section: General meaning"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>More generally, <b>direction cosine</b> refers to the cosine of the angle between any two <a href="/wiki/Euclidean_vector" title="Euclidean vector">vectors</a>. They are useful for forming <a href="/wiki/Euclidean_vector#Conversion_between_multiple_Cartesian_bases" title="Euclidean vector">direction cosine matrices</a> that express one set of <a href="/wiki/Orthonormal" class="mw-redirect" title="Orthonormal">orthonormal</a> <a href="/wiki/Basis_vectors" class="mw-redirect" title="Basis vectors">basis vectors</a> in terms of another set, or for expressing a known <a href="/wiki/Euclidean_vector" title="Euclidean vector">vector</a> in a different basis. Simply put, direction cosines provide an easy method of representing the direction of a vector in a Cartesian coordinate system. </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Direction_cosine&action=edit&section=3" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Determining_angles_between_two_vectors">Determining angles between two vectors</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Direction_cosine&action=edit&section=4" title="Edit section: Determining angles between two vectors"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If vectors <span class="texhtml"><b>u</b></span> and <span class="texhtml"><b>v</b></span> have direction cosines <span class="texhtml">(<i>α<sub>u</sub></i>, <i>β<sub>u</sub></i>, <i>γ<sub>u</sub></i>)</span> and <span class="texhtml">(<i>α<sub>v</sub></i>, <i>β<sub>v</sub></i>, <i>γ<sub>v</sub></i>)</span> respectively, with an angle <span class="texhtml mvar" style="font-style:italic;">θ</span> between them, their units vectors are </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 {\begin{aligned}\mathbf {\hat {u}} &={\frac {1}{\sqrt {u_{x}^{2}+u_{y}^{2}+u_{z}^{2}}}}\left(u_{x}\mathbf {e} _{x}+u_{y}\mathbf {e} _{y}+u_{z}\mathbf {e} _{z}\right)=\alpha _{u}\mathbf {e} _{x}+\beta _{u}\mathbf {e} _{y}+\gamma _{u}\mathbf {e} _{z}\\\mathbf {\hat {v}} &={\frac {1}{\sqrt {v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}}}\left(v_{x}\mathbf {e} _{x}+v_{y}\mathbf {e} _{y}+v_{z}\mathbf {e} _{z}\right)=\alpha _{v}\mathbf {e} _{x}+\beta _{v}\mathbf {e} _{y}+\gamma _{v}\mathbf {e} _{z}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">u</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <msubsup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">v</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {\hat {u}} &={\frac {1}{\sqrt {u_{x}^{2}+u_{y}^{2}+u_{z}^{2}}}}\left(u_{x}\mathbf {e} _{x}+u_{y}\mathbf {e} _{y}+u_{z}\mathbf {e} _{z}\right)=\alpha _{u}\mathbf {e} _{x}+\beta _{u}\mathbf {e} _{y}+\gamma _{u}\mathbf {e} _{z}\\\mathbf {\hat {v}} &={\frac {1}{\sqrt {v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}}}\left(v_{x}\mathbf {e} _{x}+v_{y}\mathbf {e} _{y}+v_{z}\mathbf {e} _{z}\right)=\alpha _{v}\mathbf {e} _{x}+\beta _{v}\mathbf {e} _{y}+\gamma _{v}\mathbf {e} _{z}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/890d9bfc87985ce37447ba19386d0de3510d6148" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.505ex; width:66.638ex; height:16.176ex;" alt="{\displaystyle {\begin{aligned}\mathbf {\hat {u}} &={\frac {1}{\sqrt {u_{x}^{2}+u_{y}^{2}+u_{z}^{2}}}}\left(u_{x}\mathbf {e} _{x}+u_{y}\mathbf {e} _{y}+u_{z}\mathbf {e} _{z}\right)=\alpha _{u}\mathbf {e} _{x}+\beta _{u}\mathbf {e} _{y}+\gamma _{u}\mathbf {e} _{z}\\\mathbf {\hat {v}} &={\frac {1}{\sqrt {v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}}}\left(v_{x}\mathbf {e} _{x}+v_{y}\mathbf {e} _{y}+v_{z}\mathbf {e} _{z}\right)=\alpha _{v}\mathbf {e} _{x}+\beta _{v}\mathbf {e} _{y}+\gamma _{v}\mathbf {e} _{z}.\end{aligned}}}"></span> Taking the <a href="/wiki/Dot_product" title="Dot product">dot product</a> of these two unit vectors yield, <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 {{\hat {u}}\cdot {\hat {v}}} =\alpha _{u}\alpha _{v}+\beta _{u}\beta _{v}+\gamma _{u}\gamma _{v}=\cos \theta ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">u</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">v</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mo>=</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {{\hat {u}}\cdot {\hat {v}}} =\alpha _{u}\alpha _{v}+\beta _{u}\beta _{v}+\gamma _{u}\gamma _{v}=\cos \theta ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e369b0452d5e20e5b839d6f79e4b3e7cf73b2bb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.31ex; height:2.843ex;" alt="{\displaystyle \mathbf {{\hat {u}}\cdot {\hat {v}}} =\alpha _{u}\alpha _{v}+\beta _{u}\beta _{v}+\gamma _{u}\gamma _{v}=\cos \theta ,}"></span> where <span class="texhtml mvar" style="font-style:italic;">θ</span> is the angle between the two unit vectors, and is also the angle between <span class="texhtml"><b>u</b></span> and <span class="texhtml"><b>v</b></span>. </p><p>Since <span class="texhtml mvar" style="font-style:italic;">θ</span> is a geometric angle, and is never negative. Therefore only the positive value of the dot product is taken, yielding us the final result, </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 \theta =\arccos \left(\alpha _{u}\alpha _{v}+\beta _{u}\beta _{v}+\gamma _{u}\gamma _{v}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mi>arccos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =\arccos \left(\alpha _{u}\alpha _{v}+\beta _{u}\beta _{v}+\gamma _{u}\gamma _{v}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d8ac1ede3010a08e023daf12fd69a2cc103e171" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.165ex; height:2.843ex;" alt="{\displaystyle \theta =\arccos \left(\alpha _{u}\alpha _{v}+\beta _{u}\beta _{v}+\gamma _{u}\gamma _{v}\right).}"></span> </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=Direction_cosine&action=edit&section=5" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Cartesian_tensor" title="Cartesian tensor">Cartesian tensor</a></li> <li><a href="/wiki/Euler_angles" title="Euler angles">Euler angles</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=Direction_cosine&action=edit&section=6" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><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 (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFKay1988" class="citation book cs1">Kay, D. C. (1988). <i>Tensor Calculus</i>. Schaum’s Outlines. McGraw Hill. pp. 18–19. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-07-033484-6" title="Special:BookSources/0-07-033484-6"><bdi>0-07-033484-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Tensor+Calculus&rft.series=Schaum%E2%80%99s+Outlines&rft.pages=18-19&rft.pub=McGraw+Hill&rft.date=1988&rft.isbn=0-07-033484-6&rft.aulast=Kay&rft.aufirst=D.+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirection+cosine" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSpiegelLipschutzSpellman2009" class="citation book cs1">Spiegel, M. R.; Lipschutz, S.; Spellman, D. (2009). <i>Vector analysis</i>. Schaum’s Outlines (2nd ed.). McGraw Hill. pp. 15, 25. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-161545-7" title="Special:BookSources/978-0-07-161545-7"><bdi>978-0-07-161545-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Vector+analysis&rft.series=Schaum%E2%80%99s+Outlines&rft.pages=15%2C+25&rft.edition=2nd&rft.pub=McGraw+Hill&rft.date=2009&rft.isbn=978-0-07-161545-7&rft.aulast=Spiegel&rft.aufirst=M.+R.&rft.au=Lipschutz%2C+S.&rft.au=Spellman%2C+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirection+cosine" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTyldesley1975" class="citation book cs1">Tyldesley, J. R. (1975). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=PODXAAAAMAAJ"><i>An introduction to tensor analysis for engineers and applied scientists</i></a>. Longman. p. 5. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-582-44355-5" title="Special:BookSources/0-582-44355-5"><bdi>0-582-44355-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+introduction+to+tensor+analysis+for+engineers+and+applied+scientists&rft.pages=5&rft.pub=Longman&rft.date=1975&rft.isbn=0-582-44355-5&rft.aulast=Tyldesley&rft.aufirst=J.+R.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DPODXAAAAMAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirection+cosine" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTang2006" class="citation book cs1">Tang, K. T. (2006). <i>Mathematical Methods for Engineers and Scientists</i>. Vol. 2. Springer. p. 13. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-30268-9" title="Special:BookSources/3-540-30268-9"><bdi>3-540-30268-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Methods+for+Engineers+and+Scientists&rft.pages=13&rft.pub=Springer&rft.date=2006&rft.isbn=3-540-30268-9&rft.aulast=Tang&rft.aufirst=K.+T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirection+cosine" class="Z3988"></span></li> <li><span class="citation mathworld" id="Reference-Mathworld-Direction_Cosine"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/DirectionCosine.html">"Direction Cosine"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Direction+Cosine&rft.au=Weisstein%2C+Eric+W.&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FDirectionCosine.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirection+cosine" class="Z3988"></span></span></li></ul> <p><br /> </p> <style data-mw-deduplicate="TemplateStyles:r1012311289">.mw-parser-output .asbox{position:relative;overflow:hidden}.mw-parser-output .asbox table{background:transparent}.mw-parser-output .asbox p{margin:0}.mw-parser-output .asbox p+p{margin-top:0.25em}.mw-parser-output .asbox-body{font-style:italic}.mw-parser-output .asbox-note{font-size:smaller}.mw-parser-output .asbox .navbar{position:absolute;top:-0.75em;right:1em;display:none}</style><div role="note" class="metadata plainlinks asbox stub"><table role="presentation"><tbody><tr class="noresize"><td><span typeof="mw:File"><a href="/wiki/File:Dodecahedron.svg" class="mw-file-description"><img alt="Stub icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Dodecahedron.svg/30px-Dodecahedron.svg.png" decoding="async" width="30" height="30" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Dodecahedron.svg/45px-Dodecahedron.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Dodecahedron.svg/60px-Dodecahedron.svg.png 2x" data-file-width="300" data-file-height="300" /></a></span></td><td><p class="asbox-body">This <a href="/wiki/Geometry" title="Geometry">geometry-related</a> article is a <a href="/wiki/Wikipedia:Stub" title="Wikipedia:Stub">stub</a>. 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