Vector Triple Product

1. The Vector Triple Product of 3 3-Dimensional Vectors \(\vec{A}\), \(\vec{B}\) and \(\vec{C}\) is calculated as follows
   
   \(\vec{A} \times (\vec{B} \times \vec{C}) = (\vec{A}\cdot\vec{C})\vec{B}-(\vec{A}\cdot\vec{B})\vec{C}\)
   
   \((\vec{A} \times \vec{B}) \times \vec{C} = (\vec{A}\cdot\vec{C})\vec{B}-(\vec{B}\cdot\vec{C})\vec{A}\)
2. Following gives the Derivation of Formula of Vector Triple Product
   
   \(\vec{B}\times\vec{C} \Rightarrow \begin{vmatrix} i & j & k \\ B_x & B_y & B_z \\ C_x & C_y & C_z\end{vmatrix} \)
   
   \(\Rightarrow (B_yC_z-B_zC_y)\hat{\textbf{i}} - (B_xC_z-B_zC_x)\hat{\textbf{j}} + (B_xC_y-B_yC_x)\hat{\textbf{k}}\)
   
   Then
   
   \(\vec{A} \times (\vec{B} \times \vec{C}) = \begin{vmatrix} i & j & k \\ A_x & A_y & A_z \\ (B_yC_z-B_zC_y) & (B_zC_x-B_xC_z) & (B_xC_y-B_yC_x)\end{vmatrix} \)
   
   The \(x\) component of \(\vec{A} \times (\vec{B} \times \vec{C})\) is given as
   
   \({(\vec{A} \times (\vec{B} \times \vec{C}))}_x=A_y(B_xC_y-B_yC_x) - A_z(B_zC_x-B_xC_z)=A_yB_xC_y - A_yB_yC_x - A_zB_zC_x + A_zB_xC_z \)
   
   Adding and Subtracting \(A_xB_xC_x\) to the above term we get
   
   \(A_yB_xC_y - A_yB_yC_x - A_zB_zC_x + A_zB_xC_z + A_xB_xC_x - A_xB_xC_x\)
   
   After rearranging and grouping we get the \(x\) component of \(\vec{A} \times (\vec{B} \times \vec{C})\) as
   
   \(B_x(A_xC_x + A_yC_y + A_zC_z) - C_x(A_xC_x + A_yC_y + A_zC_z)\)
   \(\Rightarrow {(\vec{A} \times (\vec{B} \times \vec{C}))}_x=(\vec{A}\cdot\vec{C})B_x - (\vec{A}\cdot\vec{B})C_x\)   ...(1)
   
   Similarly, the \(y\) and \(z\) component of \(\vec{A} \times (\vec{B} \times \vec{C})\) can are obtained as
   
   \({(\vec{A} \times (\vec{B} \times \vec{C}))}_y=(\vec{A}\cdot\vec{C})B_y - (\vec{A}\cdot\vec{B})C_y\)   ...(2)
   \({(\vec{A} \times (\vec{B} \times \vec{C}))}_z=(\vec{A}\cdot\vec{C})B_z - (\vec{A}\cdot\vec{B})C_z\)   ...(3)
   
   Adding (1), (2) & (3) we get
   
   \(\vec{A} \times (\vec{B} \times \vec{C}) = (\vec{A}\cdot\vec{C})\vec{B}-(\vec{A}\cdot\vec{B})\vec{C}\)
3. Following are Properties of Vector Triple Product
   1. For any given 3 Vectors \(\vec{A}\), \(\vec{B}\) and \(\vec{C}\) the following identity of Vector Triple Product holds true
      
      \(\vec{A} \times (\vec{B} \times \vec{C}) + \vec{C} \times (\vec{A} \times \vec{B}) + \vec{B} \times (\vec{C} \times \vec{A}) = 0\)
      
      \(\Rightarrow (\vec{A}\cdot\vec{C})\vec{B}-(\vec{A}\cdot\vec{B})\vec{C} + (\vec{C}\cdot\vec{B})\vec{A}-(\vec{C}\cdot\vec{A})\vec{B} + (\vec{B}\cdot\vec{A})\vec{C}-(\vec{B}\cdot\vec{C})\vec{A} = 0\)
      
   2. The Parenthesis in the Vector Triple Product is generally Not Interchangable. Instead the following identity is true (based on the above identity)
      
      \((\vec{A} \times \vec{B})\times\vec{C} = \vec{A} \times (\vec{B} \times \vec{C}) + \vec{B} \times (\vec{C} \times \vec{A})=(\vec{A}\cdot\vec{C})\vec{B}-(\vec{B}\cdot\vec{C})\vec{A} \)
      
   3. The Parenthesis in the Vector Triple Product is Interchangable only if 2 Vectors are Same (once again, based on the above 2 identities).
      
      \(\vec{A} \times (\vec{B}\times\vec{A}) = (\vec{A}\times\vec{B})\times\vec{A}=(\vec{A}\cdot\vec{A})\vec{B}-(\vec{A}\cdot\vec{B})\vec{A} \)
   4. If Vectors \(\vec{A}\) and \(\vec{B}\) are Perpendicular then
      
      \(\vec{A} \times (\vec{B}\times\vec{A}) = (\vec{A}\times\vec{B})\times\vec{A}=(\vec{A}\cdot\vec{A})\vec{B}\)