Vector Projection#
Revised
05 Apr 2023
Vector Projection#
Let \(\mathbf{v}\) and \(\mathbf{w}\) be two vectors with angle \(\theta\) between them.
The vector projection of \(\mathbf{v}\) onto \(\mathbf{w}\)
\( \boxed{ \begin{aligned} \mathbf{v}_{\parallel\mathbf{w}} \overset{\text{def}}{=} \text{proj}_\mathbf{w}\mathbf{v} =\frac{\mathbf{v\cdot w}}{\|\mathbf{w}\|^2}\mathbf{w} =\frac{\mathbf{v\cdot w}}{\mathbf{w\cdot w}}\mathbf{w} \end{aligned} } \)
is the component of \(\mathbf{v}\) that is parallel to \(\mathbf{w}\).
Its magnitude is given by
\( \begin{aligned} \|\text{proj}_\mathbf{w}\mathbf{v}\| &=\|\mathbf{v}\|\cos\theta &&\text{by the definition of cosine as adjacent over hypotenuse}\,\,\, \cos\theta=\frac{\|\text{proj}_\mathbf{w}\mathbf{v}\|}{\|\mathbf{v}\|} \\ &=\cancel{\|\mathbf{v}\|}\frac{\mathbf{v\cdot w}}{\cancel{\|\mathbf{v}\|}\|\mathbf{w}\|} &&\text{by the definition of the dot product}\,\,\, \mathbf{v\cdot w}=\|\mathbf{v}\|\|\mathbf{w}\|\cos\theta \\ &=\frac{\mathbf{v\cdot w}}{\|\mathbf{w}\|} \end{aligned} \)
Its direction is given by
\( \begin{aligned} \frac{\mathbf{w}}{\|\mathbf{w}\|} &&\text{by the definition of the unit vector} \end{aligned} \)
Therefore
\( \begin{aligned} \text{proj}_\mathbf{w}\mathbf{v} &=\frac{\mathbf{v\cdot w}}{\|\mathbf{w}\|}\frac{\mathbf{w}}{\|\mathbf{w}\|} \\ &=\frac{\mathbf{v\cdot w}}{\|\mathbf{w}\|\|\mathbf{w}\|}\mathbf{w} \\ &=\frac{\mathbf{v\cdot w}}{\mathbf{w\cdot w}}\mathbf{w} &&\text{by the definition of the dot product}\,\,\, \frac{\mathbf{w\cdot w}}{\cos(0)}=\|\mathbf{w}\|\|\mathbf{w}\| \end{aligned} \)
Vector \(\mathbf{v}\) can be decomposed into orthogonal component vectors \( \mathbf{v}_{\parallel\mathbf{w}} \) and \( \mathbf{v}_{\perp\mathbf{w}} \) wrt vector \(\mathbf{w}\).
\( \mathbf{v}_{\parallel\mathbf{w}}+\mathbf{v}_{\perp\mathbf{w}}=\mathbf{v} \iff \mathbf{v}_{\perp\mathbf{w}}=\mathbf{v}-\mathbf{v}_{\parallel\mathbf{w}} \iff \mathbf{v}_{\perp\mathbf{w}}=\mathbf{v}-\text{proj}_\mathbf{w}\mathbf{v} \)
\( \mathbf{v}_{\parallel\mathbf{w}} \) is the component of \(\mathbf{v}\) parallel to \(\mathbf{w}\)
\( \mathbf{v}_{\perp\mathbf{w}} \) is the component of \(\mathbf{v}\) perpendicular to \(\mathbf{w}\)