Linear and multilinear algebra; matrix theory Proposition

The cross product is perpendicular to its factors

Builds on Cross product Vectors are perpendicular if and only if their dot product is zero

Statement

Let $a$ and $b$ be vectors in $\mathbb{R}^3$ , and let $a \times b$ be their cross product. Let $u \cdot v = u_1 v_1 + u_2 v_2 + u_3 v_3$ be the dot product. Then $a \cdot (a \times b) = 0 \quad \text{and} \quad b \cdot (a \times b) = 0.$

Remarks

By the test with the dot product, $a \times b$ is perpendicular to $a$ and to $b$ when all three vectors are nonzero. The proof is a direct computation. For example, $a \cdot (a \times b) = a_1(a_2 b_3 - a_3 b_2) + a_2(a_3 b_1 - a_1 b_3) + a_3(a_1 b_2 - a_2 b_1)$ , and the six terms cancel in pairs. If $a$ and $b$ are linearly independent, then $a \times b$ is a normal vector of the plane through $0$ that contains $a$ and $b$ .

Reading path

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Angle between two vectors concept
Cross product concept
Length of a vector concept
Geometric formula for the dot product statement
Vectors are perpendicular if and only if their dot product is zero statement
The cross product is perpendicular to its factors statement

Tag	`0T4QHJXSD4`
URI	`m2://0T4QHJXSD4`
Kind	statement, proposition
Writing	W0 stub
MSC	15A72
Status	This copy of the site has no build data. Run `m2 status 0T4QHJXSD4`.
Uses	Cross product, Vectors are perpendicular if and only if their dot product is zero

The cross product is perpendicular to its factors

Statement

Remarks

Reading path

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