Linear and multilinear algebra; matrix theory Proposition

Scalar triple product

Statement

Let $a = (a_1, a_2, a_3)$ , $b = (b_1, b_2, b_3)$ and $c = (c_1, c_2, c_3)$ be vectors in $\mathbb{R}^3$ . Let $\times$ be the cross product and let $\cdot$ be the dot product. The scalar triple product of $a$ , $b$ and $c$ is the real number $a \cdot (b \times c)$ . It is equal to the determinant of the matrix whose rows are $a$ , $b$ and $c$ : $a \cdot (b \times c) = \det \begin{pmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{pmatrix}.$

Remarks

A cyclic shift of the three vectors does not change the value: $a \cdot (b \times c) = b \cdot (c \times a) = c \cdot (a \times b)$ . A swap of two vectors changes its sign. The absolute value $|a \cdot (b \times c)|$ is the volume of the parallelepiped with the edges $a$ , $b$ and $c$ . The scalar triple product is $0$ if and only if $a$ , $b$ and $c$ are linearly dependent. This happens exactly when they lie in one plane through $0$ .

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Cross product concept
Scalar triple product statement

Tag	`059Q2XWJDC`
URI	`m2://059Q2XWJDC`
Aliases	`mixed-product`, `box-product`
Kind	statement, proposition
Writing	W0 stub
MSC	15A72
Status	This copy of the site has no build data. Run `m2 status 059Q2XWJDC`.
Uses	Cross product

Scalar triple product

Statement

Remarks

Reading path

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