Equivalence of Definitions of Scalar Projection

Theorem

The following definitions of the concept of Scalar Projection are equivalent:

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 $\mathbf v$.

Proof

\(\ds \norm {\mathbf u} \norm {\mathbf v} \cos \theta\)

\(=\)

\(\ds \mathbf u \cdot \mathbf v\)

Definition of Dot Product

\(\ds \leadsto \ \ \)

\(\ds \norm {\mathbf u} \cos \theta\)

\(=\)

\(\ds \dfrac {\mathbf u \cdot \mathbf v} {\norm {\mathbf v} }\)

Definition 1 of Scalar Projection and Definition 2 of Scalar Projection

\(\ds \)

\(=\)

\(\ds \mathbf u \cdot \dfrac {\mathbf v} {\norm {\mathbf v} }\)

\(\ds \)

\(=\)

\(\ds \mathbf u \cdot \mathbf {\hat v}\)

Unit Vector in Direction of Vector, hence Definition 3 of Scalar Projection

$\blacksquare$