Non-Orthogonal/Oblique Vector Projection / Rejection

1. Finding the Projection and Rejection of a Vector on/from another Vector is same as Finding the Projection and Rejection of a Vector on/from a Vector Space/Sub-Space, when the Vector Sub-Space consists of Only a Single Vector.
2. As given in Fig. 1 above, the Vector \(\vec{C'_{||}}\) which is the Non-Orthogonal/Oblique Projection of Vector \(\vec{C}\) ON Vector \(\vec{A}\) is calculated as follows
   
   \(\vec{C'_{||}} = \vec{C_{||}} + (|\vec{C_{\perp}}| \tan\alpha) \hat{A}\)
   
   \(\Rightarrow \vec{C'_{||}}= \frac{(\vec{A}\cdot\vec{C})\hat{A}}{|\vec{A}|} + \frac{(|\vec{C_{\perp}}| |\vec{B_{\perp}}|)\hat{A}}{|\vec{B_{||}}|} \)
   
   \(\Rightarrow \vec{C'_{||}}= \frac{(\vec{A}\cdot\vec{C})|\vec{B_{||}}|\hat{A}+ (|\vec{C_{\perp}}| |\vec{B_{\perp}}|)|\vec{A}|\hat{A}}{|\vec{A}||\vec{B_{||}}|} \)
   
   \(\Rightarrow \vec{C'_{||}}= \frac{(\vec{A}\cdot\vec{C})\vec{B_{||}}+ ( |\vec{B_{\perp}}| |\vec{C}|\cos (90^\circ - \theta))\vec{A}}{|\vec{A}||\vec{B}| \cos \alpha} \)
   
   \(\Rightarrow \vec{C'_{||}}= \frac{(\vec{B_{||}}\cdot\vec{C})\vec{A} + ( \vec{B_{\perp}}\cdot\vec{C})\vec{A}}{\vec{B}\cdot\vec{A}} \)
   
   \(\Rightarrow \vec{C'_{||}}= \frac{ ( (\vec{B_{||}} + \vec{B_{\perp}} ) \cdot\vec{C}) \vec{A}} {\vec{B}\cdot\vec{A}}= \frac{ (\vec{B}\cdot\vec{C})\vec{A}}{\vec{B}\cdot\vec{A}}= \frac{AB^T}{B^TA}C \)
   
   The Matrix \(\frac{AB^T}{B^TA}\) is the Projection Matrix for Non-Orthogonal/Oblique Projection ON Vector \(\vec{A}\) where Vector \(\vec{B}\) is Perpendicular/Orthogonal to the Non-Orthogonal Rejection Vector \(\vec{C'_R}\). Please note that when \(\hat{B}= \pm \hat{A}\) then \(\vec{C'_R} =\vec{C_{\perp}}\) (and \(\vec{C'_{||}} =\vec{C_{||}}\)) and this Projection Matrix is same as the Projection Matrix for Orthogonal Projection ON Vector \(\vec{A}\).
3. As given in Fig. 1 above, the Vector \(\vec{C'_R}\) which is the Non-Orthogonal/Oblique Rejection of Vector \(\vec{C}\) FROM Vector \(\vec{A}\) is calculated as follows
   
   \(\vec{C}=\vec{C'_{||}} + \vec{C'_R}\)
   
   \(\Rightarrow \vec{C'_R}=\vec{C}-\vec{C'_{||}}\)
   
   \(\Rightarrow \vec{C'_R}=\vec{C}-\frac{ (\vec{B}\cdot\vec{C})\vec{A}}{\vec{B}\cdot\vec{A}}=\frac{(\vec{B}\cdot\vec{A})\vec{C}-(\vec{B}\cdot\vec{C})\vec{A}}{\vec{B}\cdot\vec{A}}=\frac{\vec{B} \times(\vec{C}\times\vec{A})}{\vec{B}\cdot\vec{A}}=\frac{(\vec{A} \times \vec{C})\times\vec{B}}{\vec{B}\cdot\vec{A}}=C-\frac{AB^T}{B^TA}C=(I-\frac{AB^T}{B^TA})C\)
   
   The Matrix \(I- \frac{AB^T}{B^TA}\) is the Rejection Matrix for Non-Orthogonal/Oblique Rejection FROM Vector \(\vec{A}\). Please note that when \(\hat{B}= \pm \hat{A}\) this Rejection Matrix is same as the Rejection Matrix for Orthogonal Rejection FROM Vector \(\vec{A}\).