Vector Product

(occasionally <span style="margin: 0px; padding: 0px; border: 0px; font-variant-numeric: inherit; font-variant-east-asian: inherit; font-weight: 700; font-stretch: inherit; line-height: inherit; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Lato, Helvetica, Arial, sans-serif; vertical-align: baseline;">directed area product to emphasize the geometric significance) is a <a href="https://en.m.wikipedia.org/wiki/Binary_operation" title="Binary operation" style="margin: 0px; padding: 0px; border: 0px; font-variant-numeric: inherit; font-variant-east-asian: inherit; font-stretch: inherit; line-height: inherit; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Lato, Helvetica, Arial, sans-serif; vertical-align: baseline; color: rgb(107, 75, 161);">binary operation</a>on two <a href="https://en.m.wikipedia.org/wiki/Euclidean_vector" title="Euclidean vector" style="margin: 0px; padding: 0px; border: 0px; font-variant-numeric: inherit; font-variant-east-asian: inherit; font-stretch: inherit; line-height: inherit; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Lato, Helvetica, Arial, sans-serif; vertical-align: baseline; color: rgb(107, 75, 161);">vectors</a> in <a href="https://en.m.wikipedia.org/wiki/Three-dimensional_space" title="Three-dimensional space" style="margin: 0px; padding: 0px; border: 0px; font-variant-numeric: inherit; font-variant-east-asian: inherit; font-stretch: inherit; line-height: inherit; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Lato, Helvetica, Arial, sans-serif; vertical-align: baseline; color: rgb(107, 75, 161);">three-dimensional space</a> <span class="mwe-math-element" style="margin: 0px; padding: 0px; border: 0px; font-variant-numeric: inherit; font-variant-east-asian: inherit; font-stretch: inherit; line-height: inherit; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Lato, Helvetica, Arial, sans-serif; vertical-align: baseline;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="margin: 0px; padding: 0px; border: 0px; font: inherit; vertical-align: baseline; display: none; clip: rect(1px, 1px, 1px, 1px); overflow: hidden; position: absolute; width: 1px; height: 1px; opacity: 0;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\mathbb {R} ^{3}\right)}"><semantics><annotation encoding="application/x-tex">{\displaystyle \left(\mathbb {R} ^{3}\right)}</annotation></semantics></math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a8f66a588b6bc772326ce538a3ce07242d048c1" class="mwe-math-fallback-image-inline" aria-hidden="true" style="margin: 0px; padding: 0px; border: 0px; font: inherit; vertical-align: -1.005ex; display: inline-block; width: 4.862ex; height: 3.343ex;"><span style="margin: 0px; padding: 0px; border: 0px; font-variant-numeric: inherit; font-variant-east-asian: inherit; font-stretch: inherit; line-height: inherit; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Lato, Helvetica, Arial, sans-serif; vertical-align: baseline;"> and is denoted by the symbol <span class="mwe-math-element" style="margin: 0px; padding: 0px; border: 0px; font-variant-numeric: inherit; font-variant-east-asian: inherit; font-stretch: inherit; line-height: inherit; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Lato, Helvetica, Arial, sans-serif; vertical-align: baseline;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="margin: 0px; padding: 0px; border: 0px; font: inherit; vertical-align: baseline; display: none; clip: rect(1px, 1px, 1px, 1px); overflow: hidden; position: absolute; width: 1px; height: 1px; opacity: 0;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \times }"><semantics><annotation encoding="application/x-tex">{\displaystyle \times }</annotation></semantics></math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ffafff1ad26cbe49045f19a67ce532116a32703" class="mwe-math-fallback-image-inline" aria-hidden="true" style="margin: 0px 0px -0.19ex; padding: 0px; border: 0px; font: inherit; vertical-align: 0.019ex; display: inline-block; width: 1.808ex; height: 1.509ex;"><span style="margin: 0px; padding: 0px; border: 0px; font-variant-numeric: inherit; font-variant-east-asian: inherit; font-stretch: inherit; line-height: inherit; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Lato, Helvetica, Arial, sans-serif; vertical-align: baseline;"> . Given two <a href="https://en.m.wikipedia.org/wiki/Linearly_independent_vectors" title="Linearly independent vectors" class="mw-redirect" style="margin: 0px; padding: 0px; border: 0px; font-variant-numeric: inherit; font-variant-east-asian: inherit; font-stretch: inherit; line-height: inherit; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Lato, Helvetica, Arial, sans-serif; vertical-align: baseline; color: rgb(107, 75, 161);">linearly independent vectors</a> <span class="mwe-math-element" style="margin: 0px; padding: 0px; border: 0px; font-variant-numeric: inherit; font-variant-east-asian: inherit; font-stretch: inherit; line-height: inherit; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Lato, Helvetica, Arial, sans-serif; vertical-align: baseline;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="margin: 0px; padding: 0px; border: 0px; font: inherit; vertical-align: baseline; display: none; clip: rect(1px, 1px, 1px, 1px); overflow: hidden; position: absolute; width: 1px; height: 1px; opacity: 0;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} }"><semantics><annotation encoding="application/x-tex">{\displaystyle \mathbf {a} }</annotation></semantics></math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a957216653a9ee0d0133dcefd13fb75e36b8b9d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="margin: 0px; padding: 0px; border: 0px; font: inherit; vertical-align: -0.338ex; display: inline-block; width: 1.299ex; height: 1.676ex;"><span style="margin: 0px; padding: 0px; border: 0px; font-variant-numeric: inherit; font-variant-east-asian: inherit; font-stretch: inherit; line-height: inherit; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Lato, Helvetica, Arial, sans-serif; vertical-align: baseline;"> and <span class="mwe-math-element" style="margin: 0px; padding: 0px; border: 0px; font-variant-numeric: inherit; font-variant-east-asian: inherit; font-stretch: inherit; line-height: inherit; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Lato, Helvetica, Arial, sans-serif; vertical-align: baseline;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="margin: 0px; padding: 0px; border: 0px; font: inherit; vertical-align: baseline; display: none; clip: rect(1px, 1px, 1px, 1px); overflow: hidden; position: absolute; width: 1px; height: 1px; opacity: 0;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {b} }"><semantics><annotation encoding="application/x-tex">{\displaystyle \mathbf {b} }</annotation></semantics></math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13ebf4628a1adf07133a6009e4a78bdd990c6eb9" class="mwe-math-fallback-image-inline" aria-hidden="true" style="margin: 0px; padding: 0px; border: 0px; font: inherit; vertical-align: -0.338ex; display: inline-block; width: 1.485ex; height: 2.176ex;"><span style="margin: 0px; padding: 0px; border: 0px; font-variant-numeric: inherit; font-variant-east-asian: inherit; font-stretch: inherit; line-height: inherit; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Lato, Helvetica, Arial, sans-serif; vertical-align: baseline;"> , the cross product, <span class="mwe-math-element" style="margin: 0px; padding: 0px; border: 0px; font-variant-numeric: inherit; font-variant-east-asian: inherit; font-stretch: inherit; line-height: inherit; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Lato, Helvetica, Arial, sans-serif; vertical-align: baseline;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="margin: 0px; padding: 0px; border: 0px; font: inherit; vertical-align: baseline; display: none; clip: rect(1px, 1px, 1px, 1px); overflow: hidden; position: absolute; width: 1px; height: 1px; opacity: 0;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} \times \mathbf {b} }"><semantics><annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \times \mathbf {b} }</annotation></semantics></math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf612179bab874c94c2ea2b4a541479534c3dacc" class="mwe-math-fallback-image-inline" aria-hidden="true" style="margin: 0px; padding: 0px; border: 0px; font: inherit; vertical-align: -0.338ex; display: inline-block; width: 5.625ex; height: 2.176ex;"><span style="margin: 0px; padding: 0px; border: 0px; font-variant-numeric: inherit; font-variant-east-asian: inherit; font-stretch: inherit; line-height: inherit; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Lato, Helvetica, Arial, sans-serif; vertical-align: baseline;"> , is a vector that is <a href="https://en.m.wikipedia.org/wiki/Perpendicular" title="Perpendicular" style="margin: 0px; padding: 0px; border: 0px; font-variant-numeric: inherit; font-variant-east-asian: inherit; font-stretch: inherit; line-height: inherit; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Lato, Helvetica, Arial, sans-serif; vertical-align: baseline; color: rgb(107, 75, 161);">perpendicular</a> to both <span class="mwe-math-element" style="margin: 0px; padding: 0px; border: 0px; font-variant-numeric: inherit; font-variant-east-asian: inherit; font-stretch: inherit; line-height: inherit; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Lato, Helvetica, Arial, sans-serif; vertical-align: baseline;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="margin: 0px; padding: 0px; border: 0px; font: inherit; vertical-align: baseline; display: none; clip: rect(1px, 1px, 1px, 1px); overflow: hidden; position: absolute; width: 1px; height: 1px; opacity: 0;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} }"><semantics><annotation encoding="application/x-tex">{\displaystyle \mathbf {a} }</annotation></semantics></math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a957216653a9ee0d0133dcefd13fb75e36b8b9d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="margin: 0px; padding: 0px; border: 0px; font: inherit; vertical-align: -0.338ex; display: inline-block; width: 1.299ex; height: 1.676ex;"><span style="margin: 0px; padding: 0px; border: 0px; font-variant-numeric: inherit; font-variant-east-asian: inherit; font-stretch: inherit; line-height: inherit; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Lato, Helvetica, Arial, sans-serif; vertical-align: baseline;"> and <span class="mwe-math-element" style="margin: 0px; padding: 0px; border: 0px; font-variant-numeric: inherit; font-variant-east-asian: inherit; font-stretch: inherit; line-height: inherit; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Lato, Helvetica, Arial, sans-serif; vertical-align: baseline;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="margin: 0px; padding: 0px; border: 0px; font: inherit; vertical-align: baseline; display: none; clip: rect(1px, 1px, 1px, 1px); overflow: hidden; position: absolute; width: 1px; height: 1px; opacity: 0;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {b} }"><semantics><annotation encoding="application/x-tex">{\displaystyle \mathbf {b} }</annotation></semantics></math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13ebf4628a1adf07133a6009e4a78bdd990c6eb9" class="mwe-math-fallback-image-inline" aria-hidden="true" style="margin: 0px; padding: 0px; border: 0px; font: inherit; vertical-align: -0.338ex; display: inline-block; width: 1.485ex; height: 2.176ex;"><span style="margin: 0px; padding: 0px; border: 0px; font-variant-numeric: inherit; font-variant-east-asian: inherit; font-stretch: inherit; line-height: inherit; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Lato, Helvetica, Arial, sans-serif; vertical-align: baseline;"> and thus <a href="https://en.m.wikipedia.org/wiki/Normal_(geometry)" title="Normal (geometry)" style="margin: 0px; padding: 0px; border: 0px; font-variant-numeric: inherit; font-variant-east-asian: inherit; font-stretch: inherit; line-height: inherit; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Lato, Helvetica, Arial, sans-serif; vertical-align: baseline; color: rgb(107, 75, 161);">normal</a> to the plane containing them. It has many applications in mathematics, <a href="https://en.m.wikipedia.org/wiki/Physics" title="Physics" style="margin: 0px; padding: 0px; border: 0px; font-variant-numeric: inherit; font-variant-east-asian: inherit; font-stretch: inherit; line-height: inherit; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Lato, Helvetica, Arial, sans-serif; vertical-align: baseline; color: rgb(107, 75, 161);">physics</a>, <a href="https://en.m.wikipedia.org/wiki/Engineering" title="Engineering" style="margin: 0px; padding: 0px; border: 0px; font-variant-numeric: inherit; font-variant-east-asian: inherit; font-stretch: inherit; line-height: inherit; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Lato, Helvetica, Arial, sans-serif; vertical-align: baseline; color: rgb(107, 75, 161);">engineering</a>, and <a href="https://en.m.wikipedia.org/wiki/Computer_programming" title="Computer programming" style="margin: 0px; padding: 0px; border: 0px; font-variant-numeric: inherit; font-variant-east-asian: inherit; font-stretch: inherit; line-height: inherit; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Lato, Helvetica, Arial, sans-serif; vertical-align: baseline; color: rgb(107, 75, 161);">computer programming</a>. It should not be confused with the <a href="https://en.m.wikipedia.org/wiki/Dot_product" title="Dot product" style="margin: 0px; padding: 0px; border: 0px; font-variant-numeric: inherit; font-variant-east-asian: inherit; font-stretch: inherit; line-height: inherit; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Lato, Helvetica, Arial, sans-serif; vertical-align: baseline; color: rgb(107, 75, 161);">dot product</a>(projection product).