Category:Definitions/Vector Projections

This category contains definitions related to Vector Projections.
Related results can be found in Category:Vector Projections.

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

Definition 1

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 $\mathbf v$.

Definition 2

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 $\mathbf v$.

Definition 3

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

$\proj_\mathbf v \mathbf u = u_{\parallel \mathbf v} \mathbf {\hat v}$

where:

$u_{\parallel \mathbf v}$ denotes the scalar projection of $\mathbf u$ on $\mathbf v$

$\mathbf {\hat v}$ denotes the unit vector in the direction of $\mathbf v$.