Vector projection

Builds on Euclidean vector Length of a vector Vectors are perpendicular if and only if their dot product is zero

Definition

Let $n$ be a positive integer, let $a$ be a vector in $\mathbb{R}^n$ , and let $b$ be a nonzero vector in $\mathbb{R}^n$ . Let $a \cdot b = a_1 b_1 + \dots + a_n b_n$ be the dot product. The vector projection of $a$ onto $b$ is the vector $\operatorname{proj}_b a = \frac{a \cdot b}{b \cdot b} \, b.$ The scalar projection of $a$ onto $b$ is the real number $\dfrac{a \cdot b}{\|b\|}$ , where $\|b\|$ is the length of $b$ .

Remarks

The vector $a - \operatorname{proj}_b a$ is the vector rejection of $a$ from $b$ . Its dot product with $b$ is $0$ , so it is perpendicular to $b$ . Among all multiples $c b$ of $b$ , the projection is the one that makes $\|a - c b\|$ smallest. If $\theta$ is the angle between $a \neq 0$ and $b$ , then the scalar projection is $\|a\| \cos \theta$ .

Reading path

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Angle between two vectors concept
Euclidean vector 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
Vector projection definition

Tag	`0G78GP5TF1`
URI	`m2://0G78GP5TF1`
Kind	definition
Writing	W0 stub
MSC	51N20
Defines	Vector projection
Uses	Euclidean vector, Length of a vector, Vectors are perpendicular if and only if their dot product is zero

Vector projection

Definition

Remarks

Reading path

Further reading

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