Linear and multilinear algebra; matrix theory Proposition

Anticommutativity of the cross product

Statement

Let $a$ and $b$ be vectors in $\mathbb{R}^3$ , and let $\times$ be the cross product. Then $b \times a = -(a \times b).$

Remarks

With $b = a$ the statement gives $a \times a = -(a \times a)$ , thus $a \times a = 0$ for each vector $a$ . The proof is a direct computation: each component of $b \times a$ is the component of $a \times b$ with the two products in the other order. The cross product is thus not commutative. For example, $e_1 \times e_2 = e_3$ but $e_2 \times e_1 = -e_3$ .

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Cross product concept
Anticommutativity of the cross product statement

Tag	`07ZS685JJC`
URI	`m2://07ZS685JJC`
Kind	statement, proposition
Writing	W0 stub
MSC	15A72
Status	This copy of the site has no build data. Run `m2 status 07ZS685JJC`.
Uses	Cross product

Anticommutativity of the cross product

Statement

Remarks

Reading path

Further reading

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