Determinant Product in Arbitrary Non Standard Basis

1. Just like Cross Product, the Determinant Product of Vectors can be calculated for Vectors given in Any Arbitrary Non Standard Basis.
2. The following demonstrates the calculation of Determinant Product for 3 4-Dimensional Vectors \(\vec{A}\), \(\vec{B}\) and \(\vec{C}\) represented in any Arbitrary Non Standard Basis \(\vec{e_1}\), \(\vec{e_2}\), \(\vec{e_3}\) and \(\vec{e_4}\)
   
   \(\vec{A}=A_1\vec{e_1} + A_2\vec{e_2} + A_3\vec{e_3} + A_4\vec{e_4}\hspace{.5cm}\vec{B}=B_1\vec{e_1} + B_2\vec{e_2} + B_3\vec{e_3} + B_4\vec{e_4}\hspace{.5cm}\vec{C}=C_1\vec{e_1} + C_2\vec{e_2} + C_3\vec{e_3} + C_4\vec{e_4}\)
   
   \({[\vec{A}\hspace{.2cm}\vec{B}\hspace{.2cm}\vec{C}]}_D =\begin{vmatrix}\mathbf{\vec{e_1}} & \mathbf{\vec{e_2}} & \mathbf{\vec{e_3}} & \mathbf{\vec{e_4}} \\ A_1 & A_2 & A_3 & A_4 \\ B_1 & B_2 & B_3 & B_4 \\ C_1 & C_2 & C_3 & C_4\end{vmatrix}\)
   
   \(\Rightarrow {[\vec{A}\hspace{.2cm}\vec{B}\hspace{.2cm}\vec{C}]}_D=\begin{vmatrix}A_2 & A_3 & A_4 \\B_2 & B_3 & B_4 \\C_2 & C_3 & C_4\end{vmatrix}{[\mathbf{\vec{e_2}}\hspace{.2cm}\mathbf{\vec{e_3}}\hspace{.2cm}\mathbf{\vec{e_4}}]}_D - \begin{vmatrix}A_1 & A_3 & A_4 \\B_1 & B_3 & B_4 \\C_1 & C_3 & C_4\end{vmatrix}{[\mathbf{\vec{e_1}}\hspace{.2cm}\mathbf{\vec{e_3}}\hspace{.2cm}\mathbf{\vec{e_4}}]}_D + \begin{vmatrix}A_1 & A_2 & A_4 \\B_1 & B_2 & B_4 \\C_1 & C_2 & C_4\end{vmatrix}{[\mathbf{\vec{e_3}}\hspace{.2cm}\mathbf{\vec{e_4}}\hspace{.2cm}\mathbf{\vec{e_1}}]}_D - \begin{vmatrix}A_1 & A_2 & A_3 \\B_1 & B_2 & B_3 \\C_1 & C_2 & C_3\end{vmatrix}{[\mathbf{\vec{e_1}}\hspace{.2cm}\mathbf{\vec{e_2}}\hspace{.2cm}\mathbf{\vec{e_3}}]}_D\)
   
   Setting \(\mathbf{\vec{e_5}}={[\mathbf{\vec{e_2}}\hspace{.2cm}\mathbf{\vec{e_3}}\hspace{.2cm}\mathbf{\vec{e_4}}]}_D\),   \(\mathbf{\vec{e_6}}={[\mathbf{\vec{e_1}}\hspace{.2cm}\mathbf{\vec{e_3}}\hspace{.2cm}\mathbf{\vec{e_4}}]}_D\),   \(\mathbf{\vec{e_7}}={[\mathbf{\vec{e_1}}\hspace{.2cm}\mathbf{\vec{e_2}}\hspace{.2cm}\mathbf{\vec{e_4}}]}_D\)   and   \(\mathbf{\vec{e_8}}={[\mathbf{\vec{e_1}}\hspace{.2cm}\mathbf{\vec{e_2}}\hspace{.2cm}\mathbf{\vec{e_3}}]}_D\)   we get
   
   \({[\vec{A}\hspace{.2cm}\vec{B}\hspace{.2cm}\vec{C}]}_D=\begin{vmatrix}A_2 & A_3 & A_4 \\B_2 & B_3 & B_4 \\C_2 & C_3 & C_4\end{vmatrix}\mathbf{\vec{e_5}} - \begin{vmatrix}A_1 & A_3 & A_4 \\B_1 & B_3 & B_4 \\C_1 & C_3 & C_4\end{vmatrix}\mathbf{\vec{e_6}} + \begin{vmatrix}A_1 & A_2 & A_4 \\B_1 & B_2 & B_4 \\C_1 & C_2 & C_4\end{vmatrix}\mathbf{\vec{e_7}} - \begin{vmatrix}A_1 & A_2 & A_3 \\B_1 & B_2 & B_3 \\C_1 & C_2 & C_3\end{vmatrix}\mathbf{\vec{e_8}}\)
   
   The Determinant Product in Non Standard Basis of \(N-1\) Vectors of \(N\)-Dimensions (where \(N \geq 3\)) can be calculated similarly.
3. Following examples demonstrates the calculation of the Determinant Product of Vectors \(\vec{A}\), \(\vec{B}\) and \(\vec{C}\) represented in Non Standard Basis \(\vec{e_1}\), \(\vec{e_2}\), \(\vec{e_3}\) and \(\vec{e_4}\) as give below
   
   \(\vec{e_1}=\begin{bmatrix}1\\-2\\3\\4\end{bmatrix}\hspace{.5cm}\vec{e_2}=\begin{bmatrix}-3\\2\\4\\5\end{bmatrix}\hspace{.5cm}\vec{e_3}=\begin{bmatrix}5\\7\\3\\2\end{bmatrix}\hspace{.5cm}\vec{e_4}=\begin{bmatrix}-5\\1\\4\\3\end{bmatrix}\)
   
   \(\vec{A}=3\mathbf{\vec{e_1}} - 2\mathbf{\vec{e_2}} + 1\mathbf{\vec{e_3}} + 3\mathbf{\vec{e_4}}\)
   
   \(\vec{B}=-3\mathbf{\vec{e_1}} + 3\mathbf{\vec{e_2}} - 2\mathbf{\vec{e_3}} + 2\mathbf{\vec{e_4}}\)
   
   \(\vec{C}=1\mathbf{\vec{e_1}} + 5\mathbf{\vec{e_2}} + 2\mathbf{\vec{e_3}} - 2\mathbf{\vec{e_4}}\)
   
   \({[\vec{A}\hspace{.2cm}\vec{B}\hspace{.2cm}\vec{C}]}_D=\begin{vmatrix}\mathbf{\vec{e_1}} & \mathbf{\vec{e_2}} & \mathbf{\vec{e_3}} & \mathbf{\vec{e_4}} \\ 3 & -2 & 1 & 3 \\ -3 & 3 & -2 & 2 \\ 1 & 5 & 2 & -2\end{vmatrix}\)
   
   \(\Rightarrow {[\vec{A}\hspace{.2cm}\vec{B}\hspace{.2cm}\vec{C}]}_D=\begin{vmatrix}-2 & 1 & 3 \\3 & -2 & 2 \\5 & 2 & -2\end{vmatrix}{[\mathbf{\vec{e_2}}\hspace{.2cm}\mathbf{\vec{e_3}}\hspace{.2cm}\mathbf{\vec{e_4}}]}_D - \begin{vmatrix}3 & 1 & 3 \\-3 & -2 & 2 \\1 & 2 & -2\end{vmatrix}{[\mathbf{\vec{e_1}}\hspace{.2cm}\mathbf{\vec{e_3}}\hspace{.2cm}\mathbf{\vec{e_4}}]}_D + \begin{vmatrix}3 & -2 & 3 \\-3 & 3 & 2 \\1 & 5 & -2\end{vmatrix}{[\mathbf{\vec{e_3}}\hspace{.2cm}\mathbf{\vec{e_4}}\hspace{.2cm}\mathbf{\vec{e_1}}]}_D - \begin{vmatrix}3 & -2 & 1 \\-3 & 3 & -2 \\1 & 5 & 2\end{vmatrix}{[\mathbf{\vec{e_1}}\hspace{.2cm}\mathbf{\vec{e_2}}\hspace{.2cm}\mathbf{\vec{e_3}}]}_D\)
   
   Calculating the Determinants and setting \(\mathbf{\vec{e_5}}={[\mathbf{\vec{e_2}}\hspace{.2cm}\mathbf{\vec{e_3}}\hspace{.2cm}\mathbf{\vec{e_4}}]}_D\),   \(\mathbf{\vec{e_6}}={[\mathbf{\vec{e_1}}\hspace{.2cm}\mathbf{\vec{e_3}}\hspace{.2cm}\mathbf{\vec{e_4}}]}_D\),   \(\mathbf{\vec{e_7}}={[\mathbf{\vec{e_1}}\hspace{.2cm}\mathbf{\vec{e_2}}\hspace{.2cm}\mathbf{\vec{e_4}}]}_D\)   and   \(\mathbf{\vec{e_8}}={[\mathbf{\vec{e_1}}\hspace{.2cm}\mathbf{\vec{e_2}}\hspace{.2cm}\mathbf{\vec{e_3}}]}_D\)   we get
   
   \(\Rightarrow {[\vec{A}\hspace{.2cm}\vec{B}\hspace{.2cm}\vec{C}]}_D=64\mathbf{\vec{e_5}} + 16\mathbf{\vec{e_6}} - 94\mathbf{\vec{e_7}} - 22\mathbf{\vec{e_8}}\)
   
   The Basis Vectors \(\mathbf{\vec{e_5}}\), \(\mathbf{\vec{e_6}}\), \(\mathbf{\vec{e_7}}\) and \(\mathbf{\vec{e_8}}\) are calculated as follows
   
   \(\mathbf{\vec{e_5}}={[\mathbf{\vec{e_2}}\hspace{.2cm}\mathbf{\vec{e_3}}\hspace{.2cm}\mathbf{\vec{e_4}}]}_D=\begin{vmatrix}\mathbf{\hat{e_1}} & \mathbf{\hat{e_2}} & \mathbf{\hat{e_3}} & \mathbf{\hat{e_4}}\\-3 & 2 & 4 & 5 \\ 5 & 7 & 3 & 2 \\ -5 & 1 & 4 & 3\end{vmatrix}\)
   
   \(\Rightarrow \mathbf{\vec{e_5}}=\begin{vmatrix}2 & 4 & 5 \\7 & 3 & 2 \\1 & 4 & 3\end{vmatrix}\mathbf{\hat{e_1}} - \begin{vmatrix}-3 & 4 & 5 \\5 & 3 & 2 \\-5 & 4 & 3\end{vmatrix}\mathbf{\hat{e_2}} + \begin{vmatrix}-3 & 2 & 5 \\5 & 7 & 2 \\-5 & 1 & 3\end{vmatrix}\mathbf{\hat{e_3}} - \begin{vmatrix}-3 & 2 & 4 \\5 & 7 & 3 \\-5 & 1 & 4\end{vmatrix}\mathbf{\hat{e_4}}\)
   
   \(\Rightarrow \mathbf{\vec{e_5}}=51\mathbf{\hat{e_1}} - 72\mathbf{\hat{e_2}} + 93\mathbf{\hat{e_3}} - 15\mathbf{\hat{e_4}}=\begin{bmatrix}51\\-72\\93\\-15\end{bmatrix}\)
   
   \(\mathbf{\vec{e_6}}={[\mathbf{\vec{e_1}}\hspace{.2cm}\mathbf{\vec{e_3}}\hspace{.2cm}\mathbf{\vec{e_4}}]}_D=\begin{vmatrix}\mathbf{\hat{e_1}} & \mathbf{\hat{e_2}} & \mathbf{\hat{e_3}} & \mathbf{\hat{e_4}}\\1& -2 & 3 & 4 \\ 5 & 7 & 3 & 2 \\ -5 & 1 & 4 & 3\end{vmatrix}\)
   
   \(\Rightarrow \mathbf{\vec{e_6}}=\begin{vmatrix}-2 & 3 & 4 \\7 & 3 & 2 \\1 & 4 & 3\end{vmatrix}\mathbf{\hat{e_1}} - \begin{vmatrix}1 & 3 & 4 \\5 & 3 & 2 \\-5 & 4 & 3\end{vmatrix}\mathbf{\hat{e_2}} + \begin{vmatrix}1 & -2 & 4 \\5 & 7 & 2 \\-5 & 1 & 3\end{vmatrix}\mathbf{\hat{e_3}} - \begin{vmatrix}1 & -2 & 3 \\5 & 7 & 3 \\-5 & 1 & 4\end{vmatrix}\mathbf{\hat{e_4}}\)
   
   \(\Rightarrow \mathbf{\vec{e_6}}=41\mathbf{\hat{e_1}} - 66\mathbf{\hat{e_2}} + 229\mathbf{\hat{e_3}} - 215\mathbf{\hat{e_4}}=\begin{bmatrix}41\\-66\\229\\-215\end{bmatrix}\)
   
   \(\mathbf{\vec{e_7}}={[\mathbf{\vec{e_1}}\hspace{.2cm}\mathbf{\vec{e_2}}\hspace{.2cm}\mathbf{\vec{e_4}}]}_D=\begin{vmatrix}\mathbf{\hat{e_1}} & \mathbf{\hat{e_2}} & \mathbf{\hat{e_3}} & \mathbf{\hat{e_4}}\\1& -2 & 3 & 4 \\ -3 & 2 & 4 & 5 \\ -5 & 1 & 4 & 3\end{vmatrix}\)
   
   \(\Rightarrow \mathbf{\vec{e_7}}=\begin{vmatrix}-2 & 3 & 4 \\2 & 4 & 5 \\1 & 4 & 3\end{vmatrix}\mathbf{\hat{e_1}} - \begin{vmatrix}1 & 3 & 4 \\-3 & 4 & 5 \\-5 & 4 & 3\end{vmatrix}\mathbf{\hat{e_2}} + \begin{vmatrix}1 & -2 & 4 \\-3 & 2 & 5 \\-5 & 1 & 3\end{vmatrix}\mathbf{\hat{e_3}} - \begin{vmatrix}1 & -2 & 3 \\-3 & 2 & 4 \\-5 & 1 & 4\end{vmatrix}\mathbf{\hat{e_4}}\)
   
   \(\Rightarrow \mathbf{\vec{e_7}}=29\mathbf{\hat{e_1}} + 24\mathbf{\hat{e_2}} + 61\mathbf{\hat{e_3}} - 41\mathbf{\hat{e_4}}=\begin{bmatrix}29\\24\\61\\-41\end{bmatrix}\)
   
   \(\mathbf{\vec{e_8}}={[\mathbf{\vec{e_1}}\hspace{.2cm}\mathbf{\vec{e_2}}\hspace{.2cm}\mathbf{\vec{e_3}}]}_D=\begin{vmatrix}\mathbf{\hat{e_1}} & \mathbf{\hat{e_2}} & \mathbf{\hat{e_3}} & \mathbf{\hat{e_4}}\\1& -2 & 3 & 4 \\ -3 & 2 & 4 & 5 \\ 5 & 7 & 3 & 2\end{vmatrix}\)
   
   \(\Rightarrow \mathbf{\vec{e_8}}=\begin{vmatrix}-2 & 3 & 4 \\2 & 4 & 5 \\7 & 3 & 2\end{vmatrix}\mathbf{\hat{e_1}} - \begin{vmatrix}1 & 3 & 4 \\-3 & 4 & 5 \\5 & 3 & 2\end{vmatrix}\mathbf{\hat{e_2}} + \begin{vmatrix}1 & -2 & 4 \\-3 & 2 & 5 \\5 & 7 & 2\end{vmatrix}\mathbf{\hat{e_3}} - \begin{vmatrix}1 & -2 & 3 \\-3 & 2 & 4 \\5 & 7 & 3\end{vmatrix}\mathbf{\hat{e_4}}\)
   
   \(\Rightarrow \mathbf{\vec{e_8}}=19\mathbf{\hat{e_1}} + 30\mathbf{\hat{e_2}} - 217\mathbf{\hat{e_3}} + 173\mathbf{\hat{e_4}}=\begin{bmatrix}19\\30\\-217\\173\end{bmatrix}\)
   
   Here \(\mathbf{\hat{e_1}}\), \(\mathbf{\hat{e_2}}\), \(\mathbf{\hat{e_3}}\) and \(\mathbf{\hat{e_4}}\) are Identity Orthonormal Basis Vectors.