Vector projection

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 $\|a\| \cos \theta$.

Explanations

L1 human

The projection answers the question "how much of $a$ points along $b$?". For $a = (3, 4)$ and $b = (1, 0)$ the projection is $(3, 0)$. The rest, $(0, 4)$, is perpendicular to $b$.