Vectors are perpendicular if and only if their dot product is zero

Builds on Angle between two vectors Geometric formula for the dot product

Statement

Let $n$ be a positive integer, and let $u$ and $v$ be nonzero vectors in $\mathbb{R}^n$ , with dot product $u \cdot v = u_1 v_1 + \dots + u_n v_n$ . Let $\theta$ be the angle between $u$ and $v$ . Then $u \cdot v = 0$ if and only if $\theta = \pi / 2$ .

Remarks

The proof uses the geometric formula for the dot product: $u \cdot v = \|u\| \, \|v\| \cos \theta$ with $\|u\| \neq 0$ and $\|v\| \neq 0$ , and $\cos \theta = 0$ for $0 \leq \theta \leq \pi$ only when $\theta = \pi / 2$ . Two vectors $u$ and $v$ with $u \cdot v = 0$ are called orthogonal. With this word, the zero vector is orthogonal to each vector. In Mathlib the angle with the zero vector is $\pi / 2$ , so the equivalence holds there for all vectors.

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Angle between two vectors 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

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Tag	`03NK6RSRQ1`
URI	`m2://03NK6RSRQ1`
Kind	statement, proposition
Writing	W0 stub
MSC	51N20
Status	This copy of the site has no build data. Run `m2 status 03NK6RSRQ1`.
Uses	Angle between two vectors, Geometric formula for the dot product

Vectors are perpendicular if and only if their dot product is zero

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