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{excerpt}Also known as the vector product, the cross product is a way of multiplying two vectors to yield another vector.{excerpt}

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h2. Use in Physics

In mechanics, the cross product is used in calculating [torque|torque (one-dimensional)] and [angular momentum|angular momentum (one-dimensional)].  The cross product is also used in introductory electricity and magnetism.  Calculations involving the production and effects of magnetic fields generally involve the cross product.

h2. Calculating Cross Products

h4. Unit Vector Cross Products

By definition:

{latex}\begin{large}\[\hat{x}\times \hat{y}= \hat{z}\]\end{large}{latex}

and the same holds for even permutations of the order of the unit vectors, thus:

{latex}\begin{large}\[ \hat{y} \times \hat{z} = \hat{x} \]
\[ \hat{z}\times \hat{x} = \hat{y}\]\end{large}{latex}

Odd permutations reverse the sign:

{latex}\begin{large}\[ \hat{y}\times\hat{x} = -\hat{z}\]
\[\hat{z}\times\hat{y} = -\hat{x}\]
\[\hat{x}\times\hat{z} = -\hat{y}\]\end{large}{latex}

and the cross product of any vector with itself is zero:

{latex}\begin{large}\[ \hat{x}\times\hat{x} = 0\]
\[\hat{y}\times\hat{y} = 0\]
\[\hat{z}\times\hat{z} = 0\]\end{large}{latex}

{note}Note that reversing the order of the two vectors being multiplied switches the sign of the result.{note}

Using this definition, it is possible to find the componentwise cross product of two vectors:

{latex}\begin{large}\[\vec{A}\times\vec{B}=(A_{x}\hat{x}+A_{y}\hat{y}+A_{z}\hat{z})\times(B_{x}\hat{x}+B_{y}\hat{y}+B_{z}\hat{z}) = A_{x}B_{x}\hat{x}\times\hat{x} + A_{x}B_{y}\hat{x}\times\hat{y} + A_{x}B_{z}\hat{x}\times\hat{z} + A_{y}B_{x}\hat{y}\times\hat{x} +A_{y}B_{y}\hat{y}\times\hat{y}+A_{y}B_{z}\hat{y}\times\hat{z}+A_{z}B_{x}\hat{z}\times\hat{x}+A_{z}B_{y}\hat{z}\times\hat{y} + A_{z}B_{z}\hat{z}\times\hat{z}\]\end{large}{latex}

Using the relationships given above for the cross product of unit vectors, we have:

{latex}\begin{large}\[ A_{x}B_{y}\hat{z} - A_{x}B_{z}\hat{y}-A_{y}B_{x}\hat{z}+A_{y}B_{z}\hat{x} + A_{z}B_{x}\hat{y}-A_{z}B_{y}\hat{x} = (A_{y}B_{z}-A_{z}B_{y})\hat{x} + (A_{z}B_{x} - A_{x}B_{z})\hat{y} +(A_{x}B_{y}-A_{y}B_{x})\hat{z}\]\end{large}{latex}

h4. Shortcut Using Matrix Determinant

One way to remember the formula derived in the section above is to use a matrix determinant: