Definition:Vector Projection/Definition 3

Definition

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

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 $\mathbf v$.

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:

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

$\mathbf u_1$