Linear and multilinear algebra; matrix theory Proposition

Vector triple product

Statement

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

Remarks

The formula is easy to remember as "BAC minus CAB": $a \times (b \times c) = b(a \cdot c) - c(a \cdot b)$ . The cross product is not associative. The formula gives $(a \times b) \times c = (a \cdot c) \, b - (b \cdot c) \, a$ , which is in general different from $a \times (b \times c)$ . A consequence is the Jacobi identity $a \times (b \times c) + b \times (c \times a) + c \times (a \times b) = 0$ .

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Cross product concept
Vector triple product statement

Tag	`03FTB26QYD`
URI	`m2://03FTB26QYD`
Aliases	`triple-product-expansion`, `bac-cab-identity`
Kind	statement, proposition
Writing	W1 stated
MSC	15A72
Status	This copy of the site has no build data. Run `m2 status 03FTB26QYD`.
Uses	Cross product

Vector triple product

Statement

Remarks

Reading path

Further reading

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