# Definition:Vector Projection

## Definition

Let $\mathbf u$ and $\mathbf v$ be vector quantities.

### Definition 1

The (vector) projection of $\mathbf u$ onto $\mathbf v$, denoted $\proj_\mathbf v \mathbf u$, is the orthogonal projection of $\mathbf u$ onto a straight line which is parallel to $\mathbf v$.

Hence $\proj_\mathbf v \mathbf u$ is a like vector to $\mathbf v$ whose length is $\norm {\mathbf u} \cos \theta$, where:

$\norm {\mathbf u}$ is the magnitude of $\mathbf u$
$\cos \theta$ is the angle between $\mathbf u$ and $\mathbf v$.

### Definition 2

The (vector) projection of $\mathbf u$ onto $\mathbf v$ is defined and denoted:

$\proj_\mathbf v \mathbf u = \dfrac {\mathbf u \cdot \mathbf v} {\norm {\mathbf v}^2} \mathbf v$

where:

$\cdot$ denotes the dot product
$\norm {\mathbf v}$ denotes the magnitude of $\mathbf v$.

### Definition 3

The (vector) projection of $\mathbf u$ onto $\mathbf v$ is defined and denoted:

$\proj_\mathbf v \mathbf u = u_{\parallel \mathbf v} \mathbf {\hat v}$

where:

$u_{\parallel \mathbf v}$ denotes the scalar projection of $\mathbf u$ on $\mathbf v$
$\mathbf {\hat v}$ denotes the unit vector in the direction of $\mathbf v$.

## Also known as

The vector projection of $\mathbf u$ onto $\mathbf v$ is also known as:

the vector component
the vector resolution
the vector resolute

of $\mathbf u$ in the direction of $\mathbf v$.

The notation for $\proj_\mathbf v \mathbf u$ also varies throughout the literature.

The following forms can sometimes be seen:

$\mathbf u_{\parallel \mathbf v}$
$\mathbf u_1$