Orthogonal 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 Basis Vector Matrix, when the Basis Vector Matrix consists of Only a Single Vector.
2. Any Vector \(\vec{V}\) can be presented as a Sum of its Projection Vector (\(\vec{V_P}\)) and Rejection Vector (\(\vec{V_R}\)) on Any Other Vector. That is
   
   \(\vec{V} = \vec{V_P} + \vec{V_R} \)
3. When the Vector \(\vec{V}\)'s Projection and Rejection Vectors are Perpendicular to each other then Projection Vector (\(\vec{V_P}\)) is also called its Horizontal Component (\(\vec{V_H}\)) and Rejection Vector (\(\vec{V_R}\)) is also called its Vertical Component (\(\vec{V_V}\)). Hence, any Vector \(\vec{V}\) can be presented as a Sum of its Horizontal (\(\vec{V_H}\)) and Vertical (\(\vec{V_V}\)) Components Based on Any Other Vector. That is
   
   \(\vec{V} = \vec{V_H} + \vec{V_V} \)
4. The Horizontal Component (\(\vec{V_H}\)) of Vector \(\vec{V}\) is also known as its Orthogonal Parallel Component or Tangential Component or Orthogonal Projection ON the Other Vector . Its also denoted as \(\vec{V_{||}}\)
5. The Vertical Component (\(\vec{V_V}\)) of Vector \(\vec{V}\) is also known as its Perpendicular Component or Normal Component or Orthogonal Rejection FROM the Other Vector. Its also denoted as \(\vec{V_{\perp}}\)
6. Given 2 vectors \(\vec{A}\) and \(\vec{B}\), Orthogonal Projection of Vector \(\vec{B}\) ON Vector \(\vec{A}\) (\(B_H\) or \(B_{||}\)) (as given in Fig. 1 below) is calculated as follows
   
   
   From Fig. 1 we have
   
   \(\cos(\theta)=\frac{|\vec{B_H}|}{|\vec{B}|}=\frac{|\vec{B_{||}}|}{|\vec{B}|}\hspace{.6cm}\Rightarrow |\vec{B_H}|=|\vec{B_{||}}|=|\vec{B}| \cos(\theta)=\frac{|\vec{A}||\vec{B}| \cos(\theta)}{|\vec{A}|}=\frac{(\vec{A}\cdot\vec{B})}{|\vec{A}|}\)   (Length of the Projected Vector or Projection)...(1)
   
   Multiplying equation (1) with \(\hat{A}\) (Unit Vector in the Direction of \(\vec{A}\)) we get
   
   \(\hat{A}|\vec{B_H}|=\hat{A}|\vec{B_{||}}|=\vec{B_H} = \vec{B_{||}} = \frac{(\vec{A}\cdot\vec{B})\hat{A}}{|\vec{A}|} = \frac{(\vec{A}\cdot\vec{B})\vec{A}}{{|\vec{A}|}^2} = \frac{(\vec{A}\cdot\vec{B})}{\vec{A}} = (\hat{A}\cdot\vec{B})\hat{A}=\frac{A A^T B}{A^TA}=\hat{A}\hat{A}^TB\)   ...(2)
   
   The Matrix \(\hat{A}\hat{A}^T\) is the Orthogonal Projection Matrix for Projection of Any Vector ON Vector \(\vec{A}\)
   
   If \(|\vec{A}|= |\vec{B}|\) Then \(\vec{B_H} = \vec{B_{||}} = \vec{A} \cos(\theta)\)  (\(\because \cos(\theta)=\frac{(\vec{A}\cdot\vec{B})}{|\vec{A}||\vec{B}|}=\frac{(\vec{A}\cdot\vec{B})}{{|\vec{A}|}^2}\))   ...(3)
   
   Similarly, Orthogonal Projection of Vector \(\vec{A}\) ON Vector \(\vec{B}\) (\(A_H\) or \(A_{||}\)) (as given in Fig. 2 above) is calculated as follows
   
   \(\cos(\theta)=\frac{|\vec{A_H}|}{|\vec{A}|}=\frac{|\vec{A_{||}}|}{|\vec{A}|}\hspace{.6cm}\Rightarrow |\vec{A_H}|=|\vec{A_{||}}|=|\vec{A}| \cos(\theta)=\frac{|\vec{B}||\vec{A}| \cos(\theta)}{|\vec{B}|}=\frac{(\vec{B}\cdot\vec{A})}{|\vec{B}|}\)   (Length of the Projected Vector or Projection)...(4)
   
   Multiplying equation (4) with with \(\hat{B}\) (Unit Vector in the Direction of \(\vec{B}\)) we get
   
   \(\hat{B}|\vec{A_H}|=\hat{B}|\vec{A_{||}}|=\vec{A_H} = \vec{A_{||}} = \frac{(\vec{B}\cdot\vec{A})\hat{B}}{|\vec{B}|} = \frac{(\vec{B}\cdot\vec{A})\vec{B}}{{|\vec{B}|}^2} = \frac{(\vec{B}\cdot\vec{A})}{\vec{B}} = (\hat{B}\cdot\vec{A})\hat{B}=\frac{BB^TA}{B^TB}=\hat{B}\hat{B}^TA \)   ...(5)
   
   The Matrix \(\hat{B}\hat{B}^T\) is the Orthogonal Projection Matrix for Projection of Any Vector ON Vector \(\vec{B}\)
   
   If \(|\vec{A}|= |\vec{B}|\) Then \(\vec{A_H} = \vec{A_{||}} = \vec{B} \cos(\theta)\)  (\(\because \cos(\theta)=\frac{(\vec{A}\cdot\vec{B})}{|\vec{A}||\vec{B}|}=\frac{(\vec{A}\cdot\vec{B})}{{|\vec{B}|}^2}\))   ...(6)
7. Orthogonal Rejection of Vector\(\vec{B}\) FROM Vector\(\vec{A}\) (\(\vec{B_V}\) or \(\vec{B_{\perp}}\)) (as given in Fig. 1 above) is calculated as follows
   
   From Triangle Law of Vector Addition we know that
   
   \(\vec{B} = \vec{B_H} + \vec{B_V} = \vec{B_{||}} + \vec{B_{\perp}}\hspace{.6cm}\Rightarrow \vec{B_V}=\vec{B_{\perp}}= \vec{B} - \vec{B_H}=\vec{B} - \vec{B_{||}} \)   ...(7)
   
   Putting the value of \(\vec{B_H}\) (or \(\vec{B_{||}}\)) form equation (2) in equation (7) above we get
   
   \(\vec{B_V}=\vec{B_{\perp}} = \vec{B} - (\hat{A}\cdot\vec{B})\hat{A} = B-\hat{A}\hat{A}^TB=(I-\hat{A}\hat{A}^T)B = \vec{B} - \frac{(\vec{A}\cdot\vec{B})\vec{A}}{{|\vec{A}|}^2} = \frac{(\vec{A}\cdot\vec{A})\vec{B} - (\vec{A}\cdot\vec{B})\vec{A}}{{|\vec{A}|}^2} \)   ...(8)
   
   The Matrix \(I-\hat{A}\hat{A}^T\) is the Orthogonal Rejection Matrix for Rejection of Any Vector FROM Vector \(\vec{A}\), where \(I\) is the Identity Matrix
   
   Using Vector Triple Product Formula in equation (8) we get
   
   \(\vec{B_V} = \vec{B_{\perp}} = \frac{\vec{A}\times(\vec{B}\times\vec{A})}{{|\vec{A}|}^2} = \frac{(\vec{A}\times\vec{B})\times\vec{A}}{{|\vec{A}|}^2}\)   ...(9)
   
   If \(|\vec{A}|= |\vec{B}|\) Then \(\vec{B_V} = \vec{B_{\perp}} = (\hat{n}\times\vec{A}) \sin(\theta)\)  (\(\because \hat{n}\sin(\theta)=\frac{(\vec{A}\times\vec{B})}{{|\vec{A}|}^2}\))   ...(10)
   
   where \(\hat{n}\) is Unit Vector perpendicular to both \(\vec{A}\) and \(\vec{B}\)
   
   Also, Since \(\sin(\theta)=\frac{|\vec{B_V}|}{|\vec{B}|}=\frac{|\vec{B_{\perp}}|}{|\vec{B}|}\hspace{.6cm}\Rightarrow |\vec{B_V}|=|\vec{B_{\perp}}|=|\vec{B}| \sin(\theta)=\frac{|\vec{A}||\vec{B}| \sin(\theta)}{|\vec{A}|}=\frac{|\vec{A}\times\vec{B}|}{|\vec{A}|}\)   (Length of the Rejected Vector or Rejection)...(11)
   
   Similarly, Orthogonal Rejection of Vector\(\vec{A}\) FROM Vector \(\vec{B}\) (\(\vec{A_V}\) or \(\vec{A_{\perp}}\)) (as given in Fig. 2 above) is calculated as
   
   \(\vec{A_V} = \vec{A_{\perp}} = \vec{A} - (\hat{B}\cdot\vec{A})\hat{B} = A-\hat{B}\hat{B}^TA=(I-\hat{B}\hat{B}^T)A = \vec{A} - \frac{(\vec{B}\cdot\vec{A})\vec{B}}{{|\vec{B}|}^2} = \frac{(\vec{B}\cdot\vec{B})\vec{A} - (\vec{B}\cdot\vec{A})\vec{B}}{{|\vec{B}|}^2}= \frac{\vec{B}\times(\vec{A}\times\vec{B})}{{|\vec{B}|}^2} = \frac{(\vec{B}\times\vec{A})\times\vec{B}}{{|\vec{B}|}^2}\)   ...(12)
   
   The Matrix \(I-\hat{B}\hat{B}^T\) is the Orthogonal Rejection Matrix for Rejection of Any Vector FROM Vector \(\vec{B}\), where \(I\) is the Identity Matrix
   
   If \(|\vec{A}|= |\vec{B}|\) Then \(\vec{A_V} = \vec{A_{\perp}} = (\hat{n}\times\vec{B}) \sin(\theta)\)  (\(\because \hat{n}\sin(\theta)=\frac{(\vec{B}\times\vec{A})}{{|\vec{B}|}^2}\))   ...(13)
   
   where \(\hat{n}\) is Unit Vector perpendicular to both \(\vec{A}\) and \(\vec{B}\)
   
   Also, Since \(\sin(\theta)=\frac{|\vec{A_V}|}{|\vec{A}|}=\frac{|\vec{A_{\perp}}|}{|\vec{A}|}\hspace{.6cm}\Rightarrow |\vec{A_V}|=|\vec{A_{\perp}}|=|\vec{A}| \sin(\theta)=\frac{|\vec{B}||\vec{A}| \sin(\theta)}{|\vec{B}|}=\frac{|\vec{B}\times\vec{A}|}{|\vec{B}|}\)   (Length of the Rejected Vector or Rejection)...(14)
   
   
8. You can use the Orthogonal Projection / Rejection of Vectors Calculator to calculate Orthogonal Projection / Rejection of Vectors.