Definition:Scalar Projection
Definition
Let $\mathbf u$ and $\mathbf v$ be vector quantities.
Definition 1
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 $\mathbf v$.
Definition 2
The scalar projection of $\mathbf u$ onto $\mathbf v$ is defined and denoted:
- $u_{\parallel \mathbf v} = \dfrac {\mathbf u \cdot \mathbf v} {\norm {\mathbf v} }$
where:
- $\cdot$ denotes the dot product
- $\norm {\mathbf v}$ denotes the magnitude of $\mathbf v$.
Definition 3
The scalar projection of $\mathbf u$ onto $\mathbf v$ is defined and denoted:
- $u_{\parallel \mathbf v} = \mathbf u \cdot \mathbf {\hat v}$
where:
- $\cdot$ denotes the dot product
- $\mathbf {\hat v}$ denotes the unit vector in the direction of $\mathbf v$.
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:
- $u_1$
- $\norm {\proj_\mathbf v \mathbf u}$