Definition:Vector Projection
Definition
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 $\mathbf v$.
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:
- $\mathbf u_{\parallel \mathbf v}$
- $\mathbf u_1$