Definition:Vector Projection/Definition 2

Definition

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

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

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

where:

$\cdot$ denotes the dot product

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

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:

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

$\mathbf u_1$

Weisstein, Eric W. "Projection." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Projection.html