Geometric formula for the dot product

Builds on Angle between two vectors Length of a vector

Statement

Let $n$ be a positive integer, and let $u$ and $v$ be nonzero vectors in $\mathbb{R}^n$ . Let $u \cdot v = u_1 v_1 + \dots + u_n v_n$ be their dot product, let $\|u\|$ and $\|v\|$ be their lengths, and let $\theta$ be the angle between them. Then $u \cdot v = \|u\| \, \|v\| \cos \theta.$

Remarks

When the angle is defined by its cosine, the formula follows from the definition. The content is that this angle is the angle of elementary geometry. In the plane, write $u = \|u\| (\cos \alpha, \sin \alpha)$ and $v = \|v\| (\cos \beta, \sin \beta)$ . Then $u \cdot v = \|u\| \, \|v\| \cos(\alpha - \beta)$ by the addition formula for the cosine. The formula also follows from the law of cosines in the triangle with the vertices $0$ , $u$ and $v$ . If $u = 0$ or $v = 0$ , then both sides are $0$ for each value of $\theta$ .

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Angle between two vectors concept
Length of a vector concept
Geometric formula for the dot product statement

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Tag	`070TXS9AXM`
URI	`m2://070TXS9AXM`
Kind	statement, proposition
Writing	W0 stub
MSC	51N20
Status	This copy of the site has no build data. Run `m2 status 070TXS9AXM`.
Uses	Angle between two vectors, Length of a vector

Geometric formula for the dot product

Statement

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