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Scalar Triple Product

1. The Scalar Triple Product of 3 3-Dimensional Vectors \(\vec{A}\), \(\vec{B}\) and \(\vec{C}\) is calculated as follows
   
   \(\vec{A}\cdot(\vec{B}\times\vec{C}) = [\vec{A}\hspace{.1cm}\vec{B}\hspace{.1cm}\vec{C}]=\begin{vmatrix} A_x & A_y & A_z\\ B_x & B_y & B_z\\ C_x & C_y & C_z \end{vmatrix}=\begin{vmatrix} A_x & B_x & C_x\\ A_y & B_y & C_y\\ A_z & B_z & C_z \end{vmatrix} \)
   
   where
   
   \(\vec{A}= A_x\hat{\textbf{i}} + A_y\hat{\textbf{j}} + A_z\hat{\textbf{k}}\)
   \(\vec{B}= B_x\hat{\textbf{i}} + B_y\hat{\textbf{j}} + B_z\hat{\textbf{k}}\)
   \(\vec{C}= C_x\hat{\textbf{i}} + C_y\hat{\textbf{j}} + C_z\hat{\textbf{k}}\)
2. Following are Properties of Scalar Triple Product
   1. The Scalar Triple Product of 3 Vectors gives the Signed Volume of Parallelopiped Bound by the 3 Vectors
   2. The Value of the Scalar Triple Product Does Not Change if the Dot and the Cross Products are Interchanged as shown below
      
      \(\vec{A}\cdot(\vec{B}\times\vec{C}) = (\vec{A}\times\vec{B})\cdot\vec{C}\hspace{.6cm}\Rightarrow [\vec{A}\hspace{.1cm}\vec{B}\hspace{.1cm}\vec{C}] = [\vec{C}\hspace{.1cm}\vec{A}\hspace{.1cm}\vec{B}] \)
   3. The Value of the Scalar Triple Product Does Not Change on Cyclic Permutation of the Vectors as shown below
      
      \(\vec{A}\cdot(\vec{B}\times\vec{C}) = \vec{C}\cdot(\vec{A}\times\vec{B}) = \vec{B}\cdot(\vec{C}\times\vec{A})\)
      
      \(\Rightarrow [\vec{A}\hspace{.1cm}\vec{B}\hspace{.1cm}\vec{C}] = [\vec{C}\hspace{.1cm}\vec{A}\hspace{.1cm}\vec{B}] = [\vec{B}\hspace{.1cm}\vec{C}\hspace{.1cm}\vec{A}] \)
   4. The Sign of Value of the Scalar Triple Product Changes on Non-Cyclic Permutation of the Vectors as shown below
      
      \(\vec{A}\cdot(\vec{B}\times\vec{C}) = -\vec{A}\cdot(\vec{C}\times\vec{B}) = -\vec{C}\cdot(\vec{B}\times\vec{A}) = -\vec{B}\cdot(\vec{A}\times\vec{C})\)
      
      \(\Rightarrow [\vec{A}\hspace{.1cm}\vec{B}\hspace{.1cm}\vec{C}] = -[\vec{A}\hspace{.1cm}\vec{C}\hspace{.1cm}\vec{B}] = -[\vec{C}\hspace{.1cm}\vec{B}\hspace{.1cm}\vec{A}] = -[\vec{B}\hspace{.1cm}\vec{A}\hspace{.1cm}\vec{C}] \)
   5. The Value of Scalar Triple Product is 0 if any 2 Vectors are Parallel or Same, or if the 3 Vectors are Co-Planar (i.e. they lie on the same Plane)