Cross product

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This article/section deals with mathematical concepts appropriate for late high school or early college.

The cross product (or vector product) of two vectors in 3-space is itself a vector in 3-space and is written ${\vec {a}}\times {\vec {b}}$. The magnitude of the resulting vector is

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where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} is the angle between the two vectors.

For two vectors ${\vec {a}}=(a_{1},a_{2},a_{3})$ and ${\vec {b}}=(b_{1},b_{2},b_{3})$,

${\vec {a}}\times {\vec {b}}=(a_{2}b_{3}-a_{3}b_{2},a_{3}b_{1}-a_{1}b_{3},a_{1}b_{2}-a_{2}b_{1})$.

The direction of the cross product is normal to both of the vectors ${\vec {a}}$ and ${\vec {b}}$. Since there are two such directions the chosen one is defined by the right hand rule: with your right hand, point your fingers along the direction of the first vector and curl them towards the second vector. The direction your thumb points gives the direction of the cross product. This means that the cross product is anticommutative: ${\vec {a}}\times {\vec {b}}=-({\vec {b}}\times {\vec {a}})$.

The cross product is a convenient way to find the volume of a parallelepiped. One may simply take the cross product of two legs and then find the dot product of that vector with the remaining leg (assuming that the legs are vectors). So, a parallelepiped with sides Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{a},\vec{b}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{c}} will have a volume,Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} , of:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V = |\vec{a} \cdot (\vec{b}\times \vec{c})|.}

Application

The cross product is used to find the torque and other parameters in physics, and is a clever way of finding the perpendicular vector to a plane.

Two of Maxwell's Equations make use of the cross product.

See also

• Dot product