Definition: Vector projection and rejection
The projection of a vector \(\vec{v}\) onto a vector \(\vec{w}\) in \(\mathbb{R}^3\) is defined as \[ \projection{\vec{v}}{\vec{w}} = \frac{\dot{\vec{v}}{\vec{w}}}{\norm{\vec{w}}^2}\vec{w} \] If \(\vec{w}\) is unitary, then this simplifies to \[ \projection{\vec{v}}{\vec{w}} = \left(\dot{\vec{v}}{\vec{w}}\right)\vec{w} \] The rejection of a vector \(\vec{v}\) onto a vector \(\vec{w}\) in \(\mathbb{R}^3\) is defined as \[ \rejection{\vec{v}}{\vec{w}} = \vec{v} - \projection{\vec{v}}{\vec{w}} \]Vector projection matrix form
Claim The projection \(\projection{\vec{v}}{\vec{w}}\) of a vector \(\vec{v}\) into a unitary vector \(\hat{w}\) can be written in terms of a matrix vector product \((\widetilde{W}^2 - I)\vec{v}\).
Proof
\[\begin{aligned}
&=\projection{\vec{v}}{\vec{w}}
\end{aligned}\]
\[\begin{aligned}
&=\left(\dot{\vec{v}}{\vec{w}}\right)\vec{w}
\end{aligned}\]
\[\begin{aligned}
\(=(\dot{\vec{v}}{\vec{w}})\vec{w}\)
\end{aligned}\]
\[\begin{aligned}
&=
\left[\begin{matrix}
(v_x w_x + v_y w_y + v_z w_z)w_x\\
(v_x w_x + v_y w_y + v_z w_z)w_y\\
(v_x w_x + v_y w_y + v_z w_z)w_z
\end{matrix}\right]
\end{aligned}\]
\[\begin{aligned}
&=
\left[\begin{matrix}
v_x w_x^2 + v_y w_y w_x + v_z w_z w_x\\
v_x w_x w_y + v_y w_y^2 + v_z w_z w_y\\
v_x w_x w_z + v_y w_y w_z + v_z w_z^2
\end{matrix}\right]
\end{aligned}\]
\[\begin{aligned}
&=
\left[\begin{matrix}
w_x^2 & w_x w_y & w_x w_z\\
w_x w_y & w_y^2 & w_y w_z\\
w_x w_z & w_y w_z & w_z^2
\end{matrix}\right]
\left[\begin{matrix}
v_x\\
v_x\\
v_x
\end{matrix}\right]
\end{aligned}\]
\[\begin{aligned}
&=\widetilde{W}^2 + I)\vec{v}
\end{aligned}\]
\[\begin{aligned}
Definition of \(\widetilde{W}^2\).
\end{aligned}\]