Definition:Scalar Projection/Definition 1

Definition

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

The scalar projection of $\mathbf u$ onto $\mathbf v$, denoted $u_{\parallel \mathbf v}$, is the magnitude of the orthogonal projection of $\mathbf u$ onto a straight line which is parallel to $\mathbf v$.

Hence $u_{\parallel \mathbf v}$ is the magnitude $\norm {\mathbf u} \cos \theta$, where:

$\norm {\mathbf u}$ is the magnitude of $\mathbf u$

$\cos \theta$ is the angle between $\mathbf u$ and $\mathbf v$.

Also known as

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:

$u_1$

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

Also see

• Equivalence of Definitions of Scalar Projection