Definition:Vector Projection/Definition 1

Definition

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

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 $\mathbf v$.

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:

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

$\mathbf u_1$

Also see

• Equivalence of Definitions of Vector Projection