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 $\mathbf v$.


 * Vector-projection.png

Also see

 * Equivalence of Definitions of Vector Projection