CrossProduct - Maple Help

Student[MultivariateCalculus]

 CrossProduct
 return the cross product of two vectors

 Calling Sequence CrossProduct(u, v) u &x v

Parameters

 u, v - three-dimensional Vectors with algebraic entries

Description

 • The CrossProduct command computes the cross product of u and v.
 • The infix operator &x can also be used to compute the cross product.
 • The cross product of u and v is given by the formula $⟨{u}_{2}{v}_{3}-{u}_{3}{v}_{2},-{u}_{1}{v}_{3}+{u}_{3}{v}_{1},{u}_{1}{v}_{2}-{u}_{2}{v}_{1}⟩$.
 • If u and v are both row Vectors, a row Vector is returned. Otherwise, a column Vector is returned.

Examples

 > $\mathrm{with}\left(\mathrm{Student}:-\mathrm{MultivariateCalculus}\right):$
 > $u≔⟨a,b,c⟩$
 ${u}{≔}\left[\begin{array}{c}{a}\\ {b}\\ {c}\end{array}\right]$ (1)
 > $v≔⟨d,e,f⟩$
 ${v}{≔}\left[\begin{array}{c}{d}\\ {e}\\ {f}\end{array}\right]$ (2)
 > $\mathrm{CrossProduct}\left(u,v\right)$
 $\left[\begin{array}{c}{b}{}{f}{-}{c}{}{e}\\ {-}{a}{}{f}{+}{c}{}{d}\\ {a}{}{e}{-}{b}{}{d}\end{array}\right]$ (3)

The cross product is anticommutative.

 > $v\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&x\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}u$
 $\left[\begin{array}{c}{-}{b}{}{f}{+}{c}{}{e}\\ {a}{}{f}{-}{c}{}{d}\\ {-}{a}{}{e}{+}{b}{}{d}\end{array}\right]$ (4)

The cross product is not associative.

 > $w≔⟨g,h,i⟩$
 ${w}{≔}\left[\begin{array}{c}{g}\\ {h}\\ {i}\end{array}\right]$ (5)
 > $\mathrm{simplify}\left(\left(u\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&x\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}v\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&x\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}w\right)$
 $\left[\begin{array}{c}\left({-}{a}{}{f}{+}{c}{}{d}\right){}{i}{-}\left({a}{}{e}{-}{b}{}{d}\right){}{h}\\ \left({-}{b}{}{f}{+}{c}{}{e}\right){}{i}{+}\left({a}{}{e}{-}{b}{}{d}\right){}{g}\\ \left({a}{}{f}{-}{c}{}{d}\right){}{g}{+}\left({b}{}{f}{-}{c}{}{e}\right){}{h}\end{array}\right]$ (6)
 > $\mathrm{simplify}\left(u\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&x\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left(v\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&x\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}w\right)\right)$
 $\left[\begin{array}{c}\left({-}{b}{}{e}{-}{c}{}{f}\right){}{g}{+}\left({b}{}{h}{+}{c}{}{i}\right){}{d}\\ \left({-}{a}{}{d}{-}{c}{}{f}\right){}{h}{+}\left({a}{}{g}{+}{c}{}{i}\right){}{e}\\ \left({-}{a}{}{d}{-}{b}{}{e}\right){}{i}{+}\left({a}{}{g}{+}{b}{}{h}\right){}{f}\end{array}\right]$ (7)

Compatibility

 • The Student[MultivariateCalculus][CrossProduct] command was introduced in Maple 2016.