Definition:Scalar Projection/Definition 2

Definition

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

The scalar projection of $\mathbf u$ onto $\mathbf v$ is defined and denoted:

$u_{\parallel \mathbf v} = \dfrac {\mathbf u \cdot \mathbf v} {\norm {\mathbf v} }$

where:

$\cdot$ denotes the dot product

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

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

the scalar component

the scalar resolution

the scalar resolute

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

The notation for $u_{\parallel \mathbf v}$ also varies throughout the literature.

The following forms can sometimes be seen:

$u_1$

$\norm {\proj_\mathbf v \mathbf u}$

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