Equivalence of Definitions of Vector Projection
Theorem
The following definitions of the concept of Vector Projection are equivalent:
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 $\mathbf v$.
Proof
$(2) \iff (3)$
| \(\ds \norm {\mathbf u} \norm {\mathbf v} \cos \theta\) | \(=\) | \(\ds \mathbf u \cdot \mathbf v\) | Definition of Dot Product | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \norm {\mathbf u} \cos \theta\) | \(=\) | \(\ds \dfrac {\mathbf u \cdot \mathbf v} {\norm {\mathbf v} }\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds u_{\parallel \mathbf v}\) | Definition 2 of Scalar Projection | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds u_{\parallel \mathbf v} \mathbf {\hat v}\) | \(=\) | \(\ds \dfrac {\mathbf u \cdot \mathbf v} {\norm {\mathbf v} } \mathbf {\hat v}\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {\mathbf u \cdot \mathbf v} {\norm {\mathbf v} } \dfrac {\mathbf v} {\norm {\mathbf v} }\) | Unit Vector in Direction of Vector | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {\mathbf u \cdot \mathbf v} {\norm {\mathbf v}^2} \mathbf v\) |
$\Box$
$(1) \iff (3)$
By definition $1$ of vector projection:
- $\proj_\mathbf v \mathbf u$ is a like vector to $\mathbf v$ whose length is $\norm {\mathbf u} \cos \theta$
This is obtained by creating a vector quantity:
- $\paren {\norm {\mathbf u} \cos \theta} \mathbf {\hat v}$
where $\mathbf {\hat v}$ is the unit vector in the direction of $\mathbf v$.
But by Definition 1 of Scalar Projection:
- $u_{\parallel \mathbf v} = \norm {\mathbf u} \cos \theta$
$\blacksquare$