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Given two linearly independent vectors a and b, the cross product, a × b (read "a cross b"), is a vector that is perpendicular to both a and b, and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming. 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class="col"> <div class="text-muted"> An Entity of Type: <a href="http://dbpedia.org/ontology/MilitaryConflict">military conflict</a>, from Named Graph: <a href="http://dbpedia.org">http://dbpedia.org</a>, within Data Space: <a href="http://dbpedia.org">dbpedia.org</a> </div> </div> </div> <div class="row pt-2"> <div class="col-xs-9 col-sm-10"> In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here ), and is denoted by the symbol . Given two linearly independent vectors a and b, the cross product, a × b (read "a cross b"), is a vector that is perpendicular to both a and b, and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming. It should not be confused with the dot product (projection product). </div> <div class="col-xs-3 col-sm-2"> <a href="#" class="thumbnail"> <img src="http://commons.wikimedia.org/wiki/Special:FilePath/Cross_product_vector.svg?width=300" alt="thumbnail" class="img-fluid" /> </a> </div> </div> </div> </section>   <section> <div class="container-xl"> <div class="row"> <div class="table-responsive"> <table class="table table-hover table-sm table-light"> <thead> <tr> <th class="col-xs-3 ">Property</th> <th class="col-xs-9 px-3">Value</th> </tr> </thead> <tbody> <tr class="odd"><td class="col-2"><a class="uri" href="http://dbpedia.org/ontology/abstract">dbo:abstract</a> </td><td class="col-10 text-break"><ul> <li style="display:none;">En matemàtiques, el producte vectorial o producte extern és una operació entre dos vectors d'un espai euclidià tridimensional orientat que retorna un altre vector ortogonal als dos vectors originals. És diferent doncs, del producte escalar o producte intern que retorna un escalar. (ca)</li> <li style="display:none;">في الرياضيات، الضرب الاتجاهي (أو الضرب التقاطعي أو الجداء المتجهي أو الجداء الشعاعي) هو عملية ثنائية بين متجهين، في فضاء إقليدي ثلاثي الأبعاد، تكون نتيجتها متجه متعامد على المستوي الذي ينتمي له المتجهان طرفا هذه العملية. وهذا بخلاف الضرب القياسي الذي يكون حاصله كمية قياسية. إذا كان و متجهان بينهما زاوية، فإن حاصل الضرب الاتجاهي لهما هو: حيث هو متجه وحدة عمودي على المستوي الحاوي للمتجهين الأصليين ) و، و هو محدد المتجهين. (ar)</li> <li style="display:none;">Vektorový součin je v matematice binární operace vektorů v trojrozměrném vektorovém prostoru. Výsledkem této operace je vektor (na rozdíl od součinu skalárního, jehož výsledkem je při součinu dvou vektorů skalár). Výsledný vektor je kolmý k oběma původním vektorům. (cs)</li> <li style="display:none;">Στα μαθηματικά, το εξωτερικό γινόμενο, ή αλλιώς διανυσματικό γινόμενο είναι μια δυαδική λειτουργία, σε δύο διανύσματα στον τρισδιάστατο χώρο και παριστάνονται με το σύμβολο ×. Το γινόμενο a × b δύο γραμμικών ανεξαρτήτων διανυσμάτων a και b, είναι ένα τρίτο διάνυσμα το οποίο είναι κάθετο προς τα δύο (a και b). Επομένως το a × b είναι κάθετο προς το επίπεδο, που περιέχει τα a και b. Έχει πολλές εφαρμογές στα μαθηματικά, στην φυσική, στην μηχανική και στον προγραμματισμό. Δεν θα πρέπει να συγχέεται με το εσωτερικό γινόμενο. Αν δύο διανύσματα έχουν την ίδια κατεύθυνση (ή έχουν την ακριβώς αντίθετη μεταξύ τους, δηλαδή δεν είναι γραμμικώς ανεξάρτητα) ή ένα από τα δύο είναι το μηδενικό διάνυσμα, τότε το γινόμενο τους είναι το μηδενικό. Πιο γενικά, το μέγεθος του γινομένου ισούται με την περιοχή του παραλληλογράμμου, που τα διανύσματα ορίζουν τις πλευρές του. Συγκεκριμένα, το μέγεθος του γινομένου δύο καθέτων διανυσμάτων είναι το γινόμενο των μηκών τους. Υπάρχει το αντίθετο (π.χ., a × b = −(b × a)) και ισχύει η επιμεριστική ιδιότητα (π.χ., a × (b + c) = a × b + a × c). Όπως το εσωτερικό γινόμενο, εξαρτάται από τη μετρική του Ευκλείδιου χώρου. Σε αντίθεση όμως με το εσωτερικό γινόμενο, εξαρτάται από την επιλογή του προσανατολισμού. (el)</li> <li style="display:none;">En matematiko, la vektora produto aŭ kruca produto estas operacio sur du vektoroj en tri-dimensia eŭklida spaco, rezulto de kiu estas la alia vektoro. Kontraste, la skalara produto de du vektoroj estas skalaro. La vektora produto estas difinita nur en tridimensioj (aǔ pli ol tri, vidu la lastan paragrafon).Algebra strukturo difinita per la vektora produto estas ne asocieca.Simile al la skalara produto, ĝi dependas de la metriko de eŭklida spaco.Malsimile al la skalara produto, ĝi dependas ankaŭ de la elekto de orientiĝo. Por ajnaj elektoj de orientiĝo, la vektora produto devas esti estimata NE kiel vektoro, sed kiel . (eo)</li> <li>In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here ), and is denoted by the symbol . Given two linearly independent vectors a and b, the cross product, a × b (read "a cross b"), is a vector that is perpendicular to both a and b, and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming. It should not be confused with the dot product (projection product). If two vectors have the same direction or have the exact opposite direction from each other (that is, they are not linearly independent), or if either one has zero length, then their cross product is zero. More generally, the magnitude of the product equals the area of a parallelogram with the vectors for sides; in particular, the magnitude of the product of two perpendicular vectors is the product of their lengths. The cross product is anticommutative (that is, a × b = − b × a) and is distributive over addition (that is, a × (b + c) = a × b + a × c). The space together with the cross product is an algebra over the real numbers, which is neither commutative nor associative, but is a Lie algebra with the cross product being the Lie bracket. Like the dot product, it depends on the metric of Euclidean space, but unlike the dot product, it also depends on a choice of orientation (or "handedness") of the space (it's why an oriented space is needed). In connection with the cross product, the exterior product of vectors can be used in arbitrary dimensions (with a bivector or 2-form result) and is independent of the orientation of the space. The product can be generalized in various ways, using the orientation and metric structure just as for the traditional 3-dimensional cross product, one can, in n dimensions, take the product of n − 1 vectors to produce a vector perpendicular to all of them. But if the product is limited to non-trivial binary products with vector results, it exists only in three and seven dimensions. The cross-product in seven dimensions has undesirable properties (e.g. it fails to satisfy the Jacobi identity), however, so it is not used in mathematical physics to represent quantities such as multi-dimensional space-time. (See , below, for other dimensions.) (en)</li> <li style="display:none;">Das Kreuzprodukt, auch Vektorprodukt, vektorielles Produkt oder äußeres Produkt, ist eine Verknüpfung im dreidimensionalen euklidischen Vektorraum, die zwei Vektoren wieder einen Vektor zuordnet. Um es von anderen Produkten, insbesondere vom Skalarprodukt, zu unterscheiden, wird es im deutsch- und englischsprachigen Raum mit einem Malkreuz als Multiplikationszeichen geschrieben (vgl. Abschnitt Schreibweisen). Die Bezeichnungen Kreuzprodukt und Vektorprodukt gehen auf den Physiker Josiah Willard Gibbs zurück, die Bezeichnung äußeres Produkt wurde von Hermann Graßmann geprägt. Das Kreuzprodukt der Vektoren und ist ein Vektor, der senkrecht auf der von den beiden Vektoren aufgespannten Ebene steht und mit ihnen ein Rechtssystem bildet. Die Länge dieses Vektors entspricht dem Flächeninhalt des Parallelogramms, das von den Vektoren und aufgespannt wird. In der Physik tritt das Kreuzprodukt an vielen Stellen auf, zum Beispiel im Elektromagnetismus bei der Berechnung der Lorentzkraft oder des Poynting-Vektors. In der klassischen Mechanik wird es bei Drehgrößen wie dem Drehmoment und dem Drehimpuls oder bei Scheinkräften wie der Corioliskraft benutzt. (de)</li> <li style="display:none;">Biderketa bektoriala hiru dimentsioko bektore-espazio batean definitzen den eragiketa bitarra da. Bi bektore harturik, haiekiko norabide elkarzuta duen bektorea du emaitza, noranzkoa eskuin eskuaren arauaaren araberakoa duena eta magnitude, luzera edo modulua a eta b bektoreak osatzen duten paralelogramoaren azalera duena. Bereziki fisikan eta ingeniaritza problemetan ditu aplikazioak. Honela kalkulatzen da, determinante baten bitartez: Biderkaduraren modulua edo norma kalkulatzeko, biderkagaien normak eta bi bektoreen arteko angeluaren sinua biderkatzea nahikoa da: (eu)</li> <li style="display:none;">En matemáticas, el producto vectorial de Gibbs o producto cruz es una operación binaria entre dos vectores en un espacio tridimensional. El resultado es un vector perpendicular a los vectores que se multiplican, y por lo tanto normal al plano que los contiene. Debido a su capacidad de obtener un vector perpendicular a otros dos vectores, cuyo sentido varía de acuerdo al ángulo formado entre estos dos vectores, esta operación es aplicada con frecuencia para resolver problemas matemáticos, físicos o de ingeniería. (es)</li> <li style="display:none;">En mathématiques, et plus précisément en géométrie, le produit vectoriel est une opération vectorielle effectuée dans les espaces euclidiens orientés de dimension 3. Le formalisme utilisé actuellement est apparu en 1881 dans un manuel d'analyse vectorielle écrit par Josiah Willard Gibbs pour ses étudiants en physique. Les travaux de Hermann Günther Grassmann et William Rowan Hamilton sont à l'origine du produit vectoriel défini par Gibbs. (fr)</li> <li style="display:none;">Perkalian vektor adalah operasi perkalian dengan dua operand (objek yang dikalikan) berupa vektor. Tetapi hasil operasi ini tidak selalu adalah vektor. Terdapat tiga macam perkalian vektor, yaitu produk skalar atau perkalian titik (bahasa Inggris: dot product atau scalar product, perkalian silang (bahasa Inggris: cross product atau vector product atau directed area product) dan perkalian langsung (bahasa Inggris: direct product). (in)</li> <li style="display:none;">선형대수학에서 벡터곱(vector곱, 영어: vector product) 또는 가위곱(영어: cross product)은 수학에서 3차원 공간의 벡터들간의 이항연산의 일종이다. 연산의 결과가 스칼라인 스칼라곱과는 달리 연산의 결과가 벡터이다. 물리학의 각운동량, 로런츠 힘 등의 공식에 등장한다. (ko)</li> <li style="display:none;">外積（がいせき）は、3次元空間（3次元内積空間）において定義される、2つのベクトルから新たなベクトルを与える二項演算である。か角括弧を用いて表現する。日本（漢字文化圏）ではこの二項演算を内積に対して外積と呼ぶ。ただし、外積に対応する西洋語（ドイツ語: Äußeres Produkt、英語: Exterior algebra）には、グラスマン代数（外積代数）のウェッジ積等の意味もあるため、区別する為にクロス積（cross product）と呼ばれる。また、内積がスカラー積と呼ばれるのに対して、ベクトル積（vector product）とも呼ばれる。なお、（outer product）は直積（direct product）を意味する。以下、この二項演算をクロス積またはベクトル積と表記する。 (ja)</li> <li style="display:none;">In matematica, in particolare nel calcolo vettoriale, il prodotto vettoriale è un'operazione binaria interna tra due vettori in uno spazio euclideo tridimensionale che restituisce un altro vettore che è normale al piano formato dai vettori di partenza. Il prodotto vettoriale è indicato con il simbolo o con il simbolo . Il secondo simbolo è però anche usato per indicare il prodotto esterno (o prodotto wedge) nell'algebra di Grassmann, di Clifford e nelle forme differenziali. Storicamente, il prodotto esterno è stato definito da Grassmann circa trent'anni prima che Gibbs e Heaviside definissero il prodotto vettoriale. (it)</li> <li style="display:none;">Векторное произведение двух векторов в трёхмерном евклидовом пространстве — вектор, перпендикулярный обоим исходным векторам, длина которого численно равна площади параллелограмма, образованного исходными векторами, а выбор из двух направлений определяется так, чтобы тройка из по порядку стоящих в произведении векторов и получившегося вектора была правой. Векторное произведение коллинеарных векторов (в частности, если хотя бы один из множителей — нулевой вектор) считается равным нулевому вектору. Таким образом, для определения векторного произведения двух векторов необходимо задать ориентацию пространства, то есть сказать, какая тройка векторов является правой, а какая — левой. При этом не является обязательным задание в рассматриваемом пространстве какой-либо системы координат. В частности, при заданной ориентации пространства результат векторного произведения не зависит от того, является ли рассматриваемая система координат правой или левой. При этом формулы выражения координат векторного произведения через координаты исходных векторов в правой и левой ортонормированной прямоугольной системе координат отличаются знаком. Векторное произведение не обладает свойствами коммутативности и ассоциативности. Оно является антикоммутативным и, в отличие от скалярного произведения векторов, результат является опять вектором. Полезно для «измерения» перпендикулярности векторов — модуль векторного произведения двух векторов равен произведению их модулей, если они перпендикулярны, и уменьшается до нуля, если векторы коллинеарны. Широко используется во многих технических и физических приложениях. Например, момент импульса и сила Лоренца математически записываются в виде векторного произведения. (ru)</li> <li style="display:none;">Het kruisproduct, vectorproduct, vectorieel product, uitwendig product of uitproduct, niet te verwarren met het Engelse 'outer product', dat een tensorproduct is, van twee vectoren in drie dimensies is een vector die loodrecht staat op beide vectoren, en waarvan de grootte gelijk is aan het product van de groottes van de beide vectoren en de sinus van de hoek tussen de twee vectoren. De richting van het kruisproduct wordt vastgelegd door de kurkentrekker- of de rechterhandregel. In tegenstelling tot het inwendig product, is het kruisproduct geen scalair, maar een vector. (nl)</li> <li style="display:none;">En kryssprodukt är en form av vektorprodukt som är definierad för vissa vektorrum (över R3 och R7). Den är antikommutativ (det vill säga, a × b = −(b × a)) och är distributiv över addition (det vill säga, a × (b + c) = a × b + a × c). Kryssprodukten är en pseudovektor. (sv)</li> <li style="display:none;">Iloczyn wektorowy – działanie dwuargumentowe przyporządkowujące parze wektorów 3-wymiarowej przestrzeni euklidesowej pewien wektor tej przestrzeni. Niech i będą wektorami 3-wymiarowej przestrzeni euklidesowej z ustaloną bazą uporządkowaną Iloczyn wektorowy wektorów i określa się następująco: * jeżeli wektory i są liniowo zależne, to * jeżeli wektory i są liniowo niezależne, to gdzie 1. * jest prostopadły zarówno do i tzn. jest wektorem normalnym do płaszczyzny wyznaczonej przez i 2. * długość wektora jest równa polu powierzchni równoległoboku wyznaczonego przez wektory i 3. * układ wektorów jest zorientowany zgodnie z bazą Wynik działania w sposób istotny zależy od doboru bazy przestrzeni. W przypadku, gdy baza trójwymiarowej przestrzeni kartezjańskiej nie jest sprecyzowana, przyjmuje się za bazę kanoniczną złożoną z wektorów (pl)</li> <li style="display:none;">Ве́кторний добу́ток — білінійна, антисиметрична операція на векторах у тривимірному просторі. На відміну від скалярного добутку векторів евклідового простору, результатом векторного добутку є вектор (його також називають «векторним добутком»), а не скаляр. Векторний добуток двох векторів у тривимірному евклідовому просторі — вектор, перпендикулярний до обох вихідних векторів, довжина якого дорівнює площі паралелограма, утвореного вихідними векторами, а вибір з двох напрямків визначається так, щоб трійка з векторів-множників, узятих в такому ж порядку, як записано в добутку, і отриманого вектора була правою. Векторний добуток колінеарних векторів (зокрема, якщо хоча б один з множників — нульовий вектор) вважається рівним нульовому вектору. Таким чином, для визначення векторного добутку двох векторів необхідно задати орієнтацію простору, тобто сказати, яка трійка векторів є правою, а яка — лівою. При цьому не є обов'язковим задання у розглянутому просторі будь-якої системи координат. Зокрема, при заданій орієнтації простору результат векторного добутку не залежить від того, чи є розглядувана система координат правою, чи лівою. При цьому формули вираження координат векторного добутку через координати вихідних векторів у правій і лівій ортонормованій прямокутній системі координат відрізняються знаком. Векторний добуток не має властивості комутативності та асоціативності. Він є антикомутативним і, на відміну від скалярного добутку векторів, результат є знову вектором. Корисний для «вимірювання» перпендикулярності векторів — модуль векторного добутку двох векторів дорівнює добутку їхніх модулів, якщо вони перпендикулярні, і зменшується до нуля, якщо вектори колінеарні. Має багато технічних і фізичних застосувань. Наприклад, момент імпульсу і сила Лоренца математично записуються у вигляді векторного добутку. (uk)</li> <li style="display:none;">Em matemática, o produto vetorial é uma operação binária sobre dois vetores em um espaço vetorial tridimensional e é denotado por ×. Dados dois vetores independentes linearmente a e b, o produto vetorial a × b é um vetor perpendicular ao vetor a e ao vetor b e é a normal do plano contendo os dois vetores. Seu resultado difere do produto escalar por ser também um vetor, ao invés de um escalar. Se dois vetores possuem a mesma direção (ou têm a exata direção oposta um ao outro, ou seja, não são linearmente independentes) ou um deles é o vetor 0, seu produto vetorial é o vetor 0. Genericamente, a magnitude do produto vetorial é igual a área do paralelogramo com os dois vetores como lados do paralelogramo. Assim, a magnitude da área do paralelogramo que possui dois vetores perpendiculares como lado é o produto do seu comprimento. (pt)</li> <li style="display:none;">在数学和向量代数领域，外積（cross product）又称叉积、叉乘、向量积（vector product），是对三维空间中的两个向量的二元运算，使用符号。与点积不同，它的运算结果是向量。对于线性无关的两个向量和，它们的外积写作，是和所在平面的法线向量，与和都垂直。外积被广泛运用于数学、物理、工程学、计算机科学领域。如果两个向量方向相同或相反（即它们没有线性无关的分量），亦或任意一个的长度为零，那么它们的外积为零。推广开来，外积的模长和以这两个向量为边的平行四边形的面积相等；如果两个向量成直角，它们外积的模长即为两者长度的乘积。外积和点积一样依赖于欧几里德空间的度量，但与点积之不同的是，外积还依赖于定向或右手定則。 (zh)</li> </ul></td></tr><tr class="even"><td class="col-2"><a class="uri" href="http://dbpedia.org/ontology/thumbnail">dbo:thumbnail</a> </td><td class="col-10 text-break"><ul> <li><a class="uri" rel="dbo:thumbnail" resource="http://commons.wikimedia.org/wiki/Special:FilePath/Cross_product_vector.svg?width=300" href="http://commons.wikimedia.org/wiki/Special:FilePath/Cross_product_vector.svg?width=300">wiki-commons:Special:FilePath/Cross_product_vector.svg?width=300</a></li> </ul></td></tr><tr class="odd"><td class="col-2"><a class="uri" href="http://dbpedia.org/ontology/wikiPageExternalLink">dbo:wikiPageExternalLink</a> </td><td 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href="http://dbpedia.org/property/p">dbp:p</a> </td><td class="col-10 text-break"><ul> <li>n–1 (en)</li> </ul></td></tr><tr class="odd"><td class="col-2"><a class="uri" href="http://dbpedia.org/property/proof">dbp:proof</a> </td><td class="col-10 text-break"><ul> <li>Evaluation of the cross product gives Hence, the left hand side equals Now, for the right hand side, And its transpose is Evaluation of the right hand side gives Comparison shows that the left hand side equals the right hand side. 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</ul></td></tr><tr class="even"><td class="col-2"><a class="uri" href="http://www.w3.org/1999/02/22-rdf-syntax-ns#type">rdf:type</a> </td><td class="col-10 text-break"><ul> <li><a class="uri" rel="rdf:type" resource="http://www.w3.org/2002/07/owl#Thing" href="http://www.w3.org/2002/07/owl#Thing">owl:Thing</a></li> <li><a class="uri" rel="rdf:type" resource="http://dbpedia.org/ontology/MilitaryConflict" href="http://dbpedia.org/ontology/MilitaryConflict">dbo:MilitaryConflict</a></li> </ul></td></tr><tr class="odd"><td class="col-2"><a class="uri" href="http://www.w3.org/2000/01/rdf-schema#comment">rdfs:comment</a> </td><td class="col-10 text-break"><ul> <li style="display:none;">En matemàtiques, el producte vectorial o producte extern és una operació entre dos vectors d'un espai euclidià tridimensional orientat que retorna un altre vector ortogonal als dos vectors originals. És diferent doncs, del producte escalar o producte intern que retorna un escalar. (ca)</li> <li style="display:none;">في الرياضيات، الضرب الاتجاهي (أو الضرب التقاطعي أو الجداء المتجهي أو الجداء الشعاعي) هو عملية ثنائية بين متجهين، في فضاء إقليدي ثلاثي الأبعاد، تكون نتيجتها متجه متعامد على المستوي الذي ينتمي له المتجهان طرفا هذه العملية. وهذا بخلاف الضرب القياسي الذي يكون حاصله كمية قياسية. إذا كان و متجهان بينهما زاوية، فإن حاصل الضرب الاتجاهي لهما هو: حيث هو متجه وحدة عمودي على المستوي الحاوي للمتجهين الأصليين ) و، و هو محدد المتجهين. (ar)</li> <li style="display:none;">Vektorový součin je v matematice binární operace vektorů v trojrozměrném vektorovém prostoru. Výsledkem této operace je vektor (na rozdíl od součinu skalárního, jehož výsledkem je při součinu dvou vektorů skalár). Výsledný vektor je kolmý k oběma původním vektorům. (cs)</li> <li style="display:none;">Biderketa bektoriala hiru dimentsioko bektore-espazio batean definitzen den eragiketa bitarra da. Bi bektore harturik, haiekiko norabide elkarzuta duen bektorea du emaitza, noranzkoa eskuin eskuaren arauaaren araberakoa duena eta magnitude, luzera edo modulua a eta b bektoreak osatzen duten paralelogramoaren azalera duena. Bereziki fisikan eta ingeniaritza problemetan ditu aplikazioak. Honela kalkulatzen da, determinante baten bitartez: Biderkaduraren modulua edo norma kalkulatzeko, biderkagaien normak eta bi bektoreen arteko angeluaren sinua biderkatzea nahikoa da: (eu)</li> <li style="display:none;">En matemáticas, el producto vectorial de Gibbs o producto cruz es una operación binaria entre dos vectores en un espacio tridimensional. El resultado es un vector perpendicular a los vectores que se multiplican, y por lo tanto normal al plano que los contiene. Debido a su capacidad de obtener un vector perpendicular a otros dos vectores, cuyo sentido varía de acuerdo al ángulo formado entre estos dos vectores, esta operación es aplicada con frecuencia para resolver problemas matemáticos, físicos o de ingeniería. (es)</li> <li style="display:none;">En mathématiques, et plus précisément en géométrie, le produit vectoriel est une opération vectorielle effectuée dans les espaces euclidiens orientés de dimension 3. Le formalisme utilisé actuellement est apparu en 1881 dans un manuel d'analyse vectorielle écrit par Josiah Willard Gibbs pour ses étudiants en physique. Les travaux de Hermann Günther Grassmann et William Rowan Hamilton sont à l'origine du produit vectoriel défini par Gibbs. (fr)</li> <li style="display:none;">Perkalian vektor adalah operasi perkalian dengan dua operand (objek yang dikalikan) berupa vektor. Tetapi hasil operasi ini tidak selalu adalah vektor. Terdapat tiga macam perkalian vektor, yaitu produk skalar atau perkalian titik (bahasa Inggris: dot product atau scalar product, perkalian silang (bahasa Inggris: cross product atau vector product atau directed area product) dan perkalian langsung (bahasa Inggris: direct product). (in)</li> <li style="display:none;">선형대수학에서 벡터곱(vector곱, 영어: vector product) 또는 가위곱(영어: cross product)은 수학에서 3차원 공간의 벡터들간의 이항연산의 일종이다. 연산의 결과가 스칼라인 스칼라곱과는 달리 연산의 결과가 벡터이다. 물리학의 각운동량, 로런츠 힘 등의 공식에 등장한다. (ko)</li> <li style="display:none;">外積（がいせき）は、3次元空間（3次元内積空間）において定義される、2つのベクトルから新たなベクトルを与える二項演算である。か角括弧を用いて表現する。日本（漢字文化圏）ではこの二項演算を内積に対して外積と呼ぶ。ただし、外積に対応する西洋語（ドイツ語: Äußeres Produkt、英語: Exterior algebra）には、グラスマン代数（外積代数）のウェッジ積等の意味もあるため、区別する為にクロス積（cross product）と呼ばれる。また、内積がスカラー積と呼ばれるのに対して、ベクトル積（vector product）とも呼ばれる。なお、（outer product）は直積（direct product）を意味する。以下、この二項演算をクロス積またはベクトル積と表記する。 (ja)</li> <li style="display:none;">Het kruisproduct, vectorproduct, vectorieel product, uitwendig product of uitproduct, niet te verwarren met het Engelse 'outer product', dat een tensorproduct is, van twee vectoren in drie dimensies is een vector die loodrecht staat op beide vectoren, en waarvan de grootte gelijk is aan het product van de groottes van de beide vectoren en de sinus van de hoek tussen de twee vectoren. De richting van het kruisproduct wordt vastgelegd door de kurkentrekker- of de rechterhandregel. In tegenstelling tot het inwendig product, is het kruisproduct geen scalair, maar een vector. (nl)</li> <li style="display:none;">En kryssprodukt är en form av vektorprodukt som är definierad för vissa vektorrum (över R3 och R7). Den är antikommutativ (det vill säga, a × b = −(b × a)) och är distributiv över addition (det vill säga, a × (b + c) = a × b + a × c). Kryssprodukten är en pseudovektor. (sv)</li> <li style="display:none;">在数学和向量代数领域，外積（cross product）又称叉积、叉乘、向量积（vector product），是对三维空间中的两个向量的二元运算，使用符号。与点积不同，它的运算结果是向量。对于线性无关的两个向量和，它们的外积写作，是和所在平面的法线向量，与和都垂直。外积被广泛运用于数学、物理、工程学、计算机科学领域。如果两个向量方向相同或相反（即它们没有线性无关的分量），亦或任意一个的长度为零，那么它们的外积为零。推广开来，外积的模长和以这两个向量为边的平行四边形的面积相等；如果两个向量成直角，它们外积的模长即为两者长度的乘积。外积和点积一样依赖于欧几里德空间的度量，但与点积之不同的是，外积还依赖于定向或右手定則。 (zh)</li> <li style="display:none;">Στα μαθηματικά, το εξωτερικό γινόμενο, ή αλλιώς διανυσματικό γινόμενο είναι μια δυαδική λειτουργία, σε δύο διανύσματα στον τρισδιάστατο χώρο και παριστάνονται με το σύμβολο ×. Το γινόμενο a × b δύο γραμμικών ανεξαρτήτων διανυσμάτων a και b, είναι ένα τρίτο διάνυσμα το οποίο είναι κάθετο προς τα δύο (a και b). Επομένως το a × b είναι κάθετο προς το επίπεδο, που περιέχει τα a και b. Έχει πολλές εφαρμογές στα μαθηματικά, στην φυσική, στην μηχανική και στον προγραμματισμό. Δεν θα πρέπει να συγχέεται με το εσωτερικό γινόμενο. (el)</li> <li style="display:none;">En matematiko, la vektora produto aŭ kruca produto estas operacio sur du vektoroj en tri-dimensia eŭklida spaco, rezulto de kiu estas la alia vektoro. Kontraste, la skalara produto de du vektoroj estas skalaro. (eo)</li> <li>In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here ), and is denoted by the symbol . Given two linearly independent vectors a and b, the cross product, a × b (read "a cross b"), is a vector that is perpendicular to both a and b, and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming. It should not be confused with the dot product (projection product). (en)</li> <li style="display:none;">Das Kreuzprodukt, auch Vektorprodukt, vektorielles Produkt oder äußeres Produkt, ist eine Verknüpfung im dreidimensionalen euklidischen Vektorraum, die zwei Vektoren wieder einen Vektor zuordnet. Um es von anderen Produkten, insbesondere vom Skalarprodukt, zu unterscheiden, wird es im deutsch- und englischsprachigen Raum mit einem Malkreuz als Multiplikationszeichen geschrieben (vgl. Abschnitt Schreibweisen). Die Bezeichnungen Kreuzprodukt und Vektorprodukt gehen auf den Physiker Josiah Willard Gibbs zurück, die Bezeichnung äußeres Produkt wurde von Hermann Graßmann geprägt. (de)</li> <li style="display:none;">In matematica, in particolare nel calcolo vettoriale, il prodotto vettoriale è un'operazione binaria interna tra due vettori in uno spazio euclideo tridimensionale che restituisce un altro vettore che è normale al piano formato dai vettori di partenza. (it)</li> <li style="display:none;">Iloczyn wektorowy – działanie dwuargumentowe przyporządkowujące parze wektorów 3-wymiarowej przestrzeni euklidesowej pewien wektor tej przestrzeni. Niech i będą wektorami 3-wymiarowej przestrzeni euklidesowej z ustaloną bazą uporządkowaną Iloczyn wektorowy wektorów i określa się następująco: Wynik działania w sposób istotny zależy od doboru bazy przestrzeni. W przypadku, gdy baza trójwymiarowej przestrzeni kartezjańskiej nie jest sprecyzowana, przyjmuje się za bazę kanoniczną złożoną z wektorów (pl)</li> <li style="display:none;">Em matemática, o produto vetorial é uma operação binária sobre dois vetores em um espaço vetorial tridimensional e é denotado por ×. Dados dois vetores independentes linearmente a e b, o produto vetorial a × b é um vetor perpendicular ao vetor a e ao vetor b e é a normal do plano contendo os dois vetores. Seu resultado difere do produto escalar por ser também um vetor, ao invés de um escalar. (pt)</li> <li style="display:none;">Векторное произведение двух векторов в трёхмерном евклидовом пространстве — вектор, перпендикулярный обоим исходным векторам, длина которого численно равна площади параллелограмма, образованного исходными векторами, а выбор из двух направлений определяется так, чтобы тройка из по порядку стоящих в произведении векторов и получившегося вектора была правой. Векторное произведение коллинеарных векторов (в частности, если хотя бы один из множителей — нулевой вектор) считается равным нулевому вектору. (ru)</li> <li style="display:none;">Ве́кторний добу́ток — білінійна, антисиметрична операція на векторах у тривимірному просторі. На відміну від скалярного добутку векторів евклідового простору, результатом векторного добутку є вектор (його також називають «векторним добутком»), а не скаляр. Векторний добуток не має властивості комутативності та асоціативності. Він є антикомутативним і, на відміну від скалярного добутку векторів, результат є знову вектором. Має багато технічних і фізичних застосувань. Наприклад, момент імпульсу і сила Лоренца математично записуються у вигляді векторного добутку. (uk)</li> </ul></td></tr><tr class="even"><td class="col-2"><a class="uri" href="http://www.w3.org/2000/01/rdf-schema#label">rdfs:label</a> </td><td class="col-10 text-break"><ul> <li style="display:none;">ضرب اتجاهي (ar)</li> <li>Cross product (en)</li> <li style="display:none;">Producte vectorial (ca)</li> <li style="display:none;">Vektorový součin (cs)</li> <li style="display:none;">Kreuzprodukt (de)</li> <li style="display:none;">Διανυσματικό γινόμενο (el)</li> <li style="display:none;">Vektora produto (eo)</li> <li style="display:none;">Producto vectorial (es)</li> <li style="display:none;">Biderketa bektorial (eu)</li> <li style="display:none;">Produit vectoriel (fr)</li> <li style="display:none;">Perkalian vektor (in)</li> <li style="display:none;">Prodotto vettoriale (it)</li> <li style="display:none;">クロス積 (ja)</li> <li style="display:none;">벡터곱 (ko)</li> <li style="display:none;">Kruisproduct (nl)</li> <li style="display:none;">Produto vetorial (pt)</li> <li style="display:none;">Iloczyn wektorowy (pl)</li> <li style="display:none;">Векторное произведение (ru)</li> <li style="display:none;">Kryssprodukt (sv)</li> <li style="display:none;">Векторний добуток (uk)</li> <li style="display:none;">叉积 (zh)</li> </ul></td></tr><tr class="odd"><td class="col-2"><a class="uri" href="http://www.w3.org/2000/01/rdf-schema#seeAlso">rdfs:seeAlso</a> </td><td class="col-10 text-break"><ul> <li><a class="uri" rel="rdfs:seeAlso" resource="http://dbpedia.org/resource/Lorentz_force" href="http://dbpedia.org/resource/Lorentz_force">dbr:Lorentz_force</a></li> <li><a class="uri" rel="rdfs:seeAlso" resource="http://dbpedia.org/resource/Seven-dimensional_cross_product" href="http://dbpedia.org/resource/Seven-dimensional_cross_product">dbr:Seven-dimensional_cross_product</a></li> <li><a class="uri" rel="rdfs:seeAlso" resource="http://dbpedia.org/resource/Triple_product" href="http://dbpedia.org/resource/Triple_product">dbr:Triple_product</a></li> </ul></td></tr><tr class="even"><td class="col-2"><a class="uri" href="http://www.w3.org/2002/07/owl#sameAs">owl:sameAs</a> </td><td class="col-10 text-break"><ul> <li><a class="uri" rel="owl:sameAs" resource="http://rdf.freebase.com/ns/m.014m_h" href="http://rdf.freebase.com/ns/m.014m_h">freebase:Cross product</a></li> <li><a class="uri" rel="owl:sameAs" resource="http://www.wikidata.org/entity/Q178192" href="http://www.wikidata.org/entity/Q178192">wikidata:Cross product</a></li> <li><a class="uri" rel="owl:sameAs" resource="http://als.dbpedia.org/resource/Kreuzprodukt" href="http://als.dbpedia.org/resource/Kreuzprodukt">dbpedia-als:Cross product</a></li> <li><a class="uri" rel="owl:sameAs" resource="http://am.dbpedia.org/resource/ስፋት_ብዜት" href="http://am.dbpedia.org/resource/ስፋት_ብዜት">http://am.dbpedia.org/resource/ስፋት_ብዜት</a></li> <li><a class="uri" rel="owl:sameAs" resource="http://ar.dbpedia.org/resource/ضرب_اتجاهي" href="http://ar.dbpedia.org/resource/ضرب_اتجاهي">dbpedia-ar:Cross product</a></li> 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class="uri" rev="dbo:wikiPageWikiLink" resource="http://dbpedia.org/resource/Specific_angular_momentum" href="http://dbpedia.org/resource/Specific_angular_momentum">dbr:Specific_angular_momentum</a></li> <li><a class="uri" rev="dbo:wikiPageWikiLink" resource="http://dbpedia.org/resource/Sarrus's_scheme" href="http://dbpedia.org/resource/Sarrus's_scheme">dbr:Sarrus's_scheme</a></li> <li><a class="uri" rev="dbo:wikiPageWikiLink" resource="http://dbpedia.org/resource/Vector_Product" href="http://dbpedia.org/resource/Vector_Product">dbr:Vector_Product</a></li> <li><a class="uri" rev="dbo:wikiPageWikiLink" resource="http://dbpedia.org/resource/Vector_cross_product" href="http://dbpedia.org/resource/Vector_cross_product">dbr:Vector_cross_product</a></li> <li><a class="uri" rev="dbo:wikiPageWikiLink" resource="http://dbpedia.org/resource/Vector_product" href="http://dbpedia.org/resource/Vector_product">dbr:Vector_product</a></li> <li><a class="uri" rev="dbo:wikiPageWikiLink" resource="http://dbpedia.org/resource/Vectorial_product" href="http://dbpedia.org/resource/Vectorial_product">dbr:Vectorial_product</a></li> <li><a class="uri" rev="dbo:wikiPageWikiLink" resource="http://dbpedia.org/resource/Ccw_test" href="http://dbpedia.org/resource/Ccw_test">dbr:Ccw_test</a></li> <li><a class="uri" rev="dbo:wikiPageWikiLink" resource="http://dbpedia.org/resource/Cross-product" href="http://dbpedia.org/resource/Cross-product">dbr:Cross-product</a></li> <li><a class="uri" rev="dbo:wikiPageWikiLink" resource="http://dbpedia.org/resource/Cross_product_matrix" href="http://dbpedia.org/resource/Cross_product_matrix">dbr:Cross_product_matrix</a></li> <li><a class="uri" rev="dbo:wikiPageWikiLink" resource="http://dbpedia.org/resource/Cross_products" href="http://dbpedia.org/resource/Cross_products">dbr:Cross_products</a></li> <li><a class="uri" rev="dbo:wikiPageWikiLink" resource="http://dbpedia.org/resource/Crossproduct" href="http://dbpedia.org/resource/Crossproduct">dbr:Crossproduct</a></li> <li><a class="uri" rev="dbo:wikiPageWikiLink" resource="http://dbpedia.org/resource/Three-dimensional_cross_product" href="http://dbpedia.org/resource/Three-dimensional_cross_product">dbr:Three-dimensional_cross_product</a></li> </ul></td></tr><tr class="even"><td class="col-2">is <a class="uri" href="http://dbpedia.org/property/knownFor">dbp:knownFor</a> of</td><td class="col-10 text-break"><ul> <li><a class="uri" rev="dbp:knownFor" resource="http://dbpedia.org/resource/Josiah_Willard_Gibbs" href="http://dbpedia.org/resource/Josiah_Willard_Gibbs">dbr:Josiah_Willard_Gibbs</a></li> </ul></td></tr><tr class="odd"><td class="col-2">is <a class="uri" href="http://www.w3.org/2000/01/rdf-schema#seeAlso">rdfs:seeAlso</a> of</td><td class="col-10 text-break"><ul> <li><a class="uri" rev="rdfs:seeAlso" resource="http://dbpedia.org/resource/Comparison_of_vector_algebra_and_geometric_algebra" href="http://dbpedia.org/resource/Comparison_of_vector_algebra_and_geometric_algebra">dbr:Comparison_of_vector_algebra_and_geometric_algebra</a></li> </ul></td></tr><tr class="even"><td class="col-2">is <a class="uri" href="http://www.w3.org/2002/07/owl#differentFrom">owl:differentFrom</a> of</td><td class="col-10 text-break"><ul> <li><a class="uri" rev="owl:differentFrom" resource="http://dbpedia.org/resource/Cross-multiplication" href="http://dbpedia.org/resource/Cross-multiplication">dbr:Cross-multiplication</a></li> <li><a class="uri" rev="owl:differentFrom" resource="http://dbpedia.org/resource/Crossed_product" href="http://dbpedia.org/resource/Crossed_product">dbr:Crossed_product</a></li> </ul></td></tr><tr class="odd"><td class="col-2">is <a class="uri" href="http://xmlns.com/foaf/0.1/primaryTopic">foaf:primaryTopic</a> of</td><td class="col-10 text-break"><ul> <li><a class="uri" rev="foaf:primaryTopic" resource="http://en.wikipedia.org/wiki/Cross_product" href="http://en.wikipedia.org/wiki/Cross_product">wikipedia-en:Cross_product</a></li> </ul></td></tr> </tbody> </table> </div> </div> </div> </section>   <section> <div class="container-xl"> <div class="text-center p-4 bg-light"> <a href="https://virtuoso.openlinksw.com/" title="OpenLink Virtuoso"><img class="powered_by" src="/statics/images/virt_power_no_border.png" alt="Powered by OpenLink Virtuoso"/></a>    <a href="http://linkeddata.org/"><img alt="This material is Open Knowledge" src="/statics/images/LoDLogo.gif"/></a>     <a href="http://dbpedia.org/sparql"><img alt="W3C Semantic Web Technology" src="/statics/images/sw-sparql-blue.png"/></a>     <a href="https://opendefinition.org/"><img alt="This material is Open Knowledge" src="/statics/images/od_80x15_red_green.png"/></a>    <a href="https://validator.w3.org/check?uri=referer"> <img src="https://www.w3.org/Icons/valid-xhtml-rdfa" alt="Valid XHTML + RDFa" /> </a> This content was extracted from <a 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