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Cross Product

Definition

The cross product of two vectors $\boldsymbol{\mathrm{a} }$ and $\boldsymbol{\mathrm{b} }$, separated by an angle of $\theta$ radians, is defined as follows:

\[\boldsymbol{\mathrm{a} }\times\boldsymbol{\mathrm{b} }=\lvert\boldsymbol{\mathrm{a} }\rvert\lvert\boldsymbol{\mathrm{b} }\rvert \sin{(\theta)}\;\boldsymbol{\mathrm{\hat{n} } }\]

$\boldsymbol{\mathrm{\hat{n} } }$ is a unit vector perpendicular to $\boldsymbol{\mathrm{a} }$ and $\boldsymbol{\mathrm{b} }$ such that $\boldsymbol{\mathrm{a} }$, $\boldsymbol{\mathrm{b} }$ and $\boldsymbol{\mathrm{\hat{n} } }$ form a right-handed system.

The cross product gives a vector which is perpendicular to both $\boldsymbol{\mathrm{a}}$ and $\boldsymbol{\mathrm{b}}$

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Calculating the Cross Product

Expressing the cross product in the form of a matrix determinant allows us to calculate each component explicitly:

\begin{align} \boldsymbol{\mathrm{a} }\times\boldsymbol{\mathrm{b} } &= \begin{vmatrix} \boldsymbol{\mathrm{e} }_1 & \boldsymbol{\mathrm{e} }_2 & \boldsymbol{\mathrm{e} }_3 \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix} \\ &=(a_2b_3-a_3b_2)\boldsymbol{\mathrm{e} }_1 + (a_3b_1-a_1b_3)\boldsymbol{\mathrm{e} }_2 + (a_1b_2-a_2b_2)\boldsymbol{\mathrm{e} }_3 \end{align}

Recall that $\boldsymbol{\mathrm{e} }_1$, $\boldsymbol{\mathrm{e} }_2$ and $\boldsymbol{\mathrm{e} }_3$ form the unit basis vectors.

Properties

The cross product is anticommutative:

  • $\boldsymbol{\mathrm{b} }\times\boldsymbol{\mathrm{a} }=-\boldsymbol{\mathrm{a} }\times\boldsymbol{\mathrm{b} }.$

The cross product is distributive over addition:

  • $\boldsymbol{\mathrm{a} }\times(\boldsymbol{\mathrm{b} }+\boldsymbol{\mathrm{c} })=\boldsymbol{\mathrm{a} }\times\boldsymbol{\mathrm{b} }+\boldsymbol{\mathrm{a} }\times\boldsymbol{\mathrm{c} },$
  • $(\boldsymbol{\mathrm{b} }+\boldsymbol{\mathrm{c} })\times\boldsymbol{\mathrm{a} }=\boldsymbol{\mathrm{b} }\times\boldsymbol{\mathrm{a} }=\boldsymbol{\mathrm{c} }\times\boldsymbol{\mathrm{a} }.$

Since the angle between parallel vectors is $0$, and $\sin{0}=0$, the cross product of a vector with itself is $0$:

  • $\boldsymbol{\mathrm{a} }\times\boldsymbol{\mathrm{a} } = \boldsymbol{0}.$

By the same argument, if $\boldsymbol{\mathrm{a} }$ and $\boldsymbol{\mathrm{b} }$ are parallel vectors, then their cross product is $0$:

  • $\boldsymbol{\mathrm{a} }\times\boldsymbol{\mathrm{b} }=\boldsymbol{\mathrm{b} }\times\boldsymbol{\mathrm{a} }=\boldsymbol{0}.$

Worked Example

Example 1

Given vectors $\boldsymbol{\mathrm{a} }=(4,\,7,\,-3)$ and $\boldsymbol{\mathrm{b} }=(5,-8,11)$, calculate $\boldsymbol{\mathrm{a} }\times\boldsymbol{\mathrm{b} }$.

Solution

Recall that the cross product can be calculated by evaluating a determinant.

\begin{align} \boldsymbol{\mathrm{a} }\times\boldsymbol{\mathrm{b} } &= \begin{vmatrix} \boldsymbol{\mathrm{e} }_1 & \boldsymbol{\mathrm{e} }_2 & \boldsymbol{\mathrm{e} }_3 \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix} \\ &=(a_2b_3-a_3b_2)\boldsymbol{\mathrm{e} }_1+(a_3b_1-a_1b_3)\boldsymbol{\mathrm{e} }_2+(a_1b_2-a_2b_1). \end{align}

For the given vectors the cross product is

\begin{align} \boldsymbol{\mathrm{a} }\times\boldsymbol{\mathrm{b} } &= \begin{vmatrix} \boldsymbol{\mathrm{e} }_1 & \boldsymbol{\mathrm{e} }_2 & \boldsymbol{\mathrm{e} }_3 \\ 4 & 7 & -3 \\ 5 & -8 & 11 \end{vmatrix} \\ &=(7\cdot11-(-3)\cdot(-8))\boldsymbol{\mathrm{e} }_1+((-3)\cdot5-4\cdot11)\boldsymbol{\mathrm{e} }_2+(4\cdot(-8)-7\cdot5)\boldsymbol{\mathrm{e} }_3 \\ &=(77-24)\boldsymbol{\mathrm{e} }_1+(-15-44)\boldsymbol{\mathrm{e} }_2+(-32-35)\boldsymbol{\mathrm{e} }_3 \\ &=53\boldsymbol{\mathrm{e} }_1-59\boldsymbol{\mathrm{e} }_2-67\boldsymbol{\mathrm{e} }_3 \end{align}

Hence,

\[\boldsymbol{\mathrm{a} }\times\boldsymbol{\mathrm{b} }= 53\boldsymbol{\mathrm{e} }_1-59\boldsymbol{\mathrm{e} }_2-67\boldsymbol{\mathrm{e} }_3.\]

Or equivalently,

\[\boldsymbol{\mathrm{a} }\times\boldsymbol{\mathrm{b} }=(53,\,-59,\,-67).\]

Workbook

This workbook produced by HELM is a good revision aid, containing key points for revision and many worked examples.

Test Yourself

Test yourself: Numbas test on vectors including dot and cross product

See Also

External Resources

Whiteboard maths

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