\( % Arcus cosine. \def\acos{\cos^{-1}} % Vector projection. \def\projection#1#2{{proj_{#1}\left(#2\right)}} % Vector rejection. \def\rejection#1#2{{rej_{#1}\left(#2\right)}} % Norm. \def\norm#1{{\left\|#1\right\|}} % Cross product. \def\cross#1#2{\mathit{cross}\left(#1,#2\right)} % Dot product. \def\dot#1#2{{#1 \cdot #2}} % Magnitude. \def\mag#1{{\left|#1\right}} \def\group#1{\left(#1\right)}} \def\sbgrp#1{\left\{#1\right\}} \)

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}\]