Determinant Product of Vectors

1. Determinant Product of Vectors is an generalization of Cross Product of Vectors for 3 or higher dimensional Vectors. It is calculted Between \(N-1\) Vectors of N-Dimensions each where \(N \geq 3\).
2. The Result of the Determinant Product of \(N-1\) N-Dimentional Vectors is another N-Dimensional Vector that is Perpendicular to All the Vectors participating in Calculation of the Determinant Product. This property of Determinant Products can be used to find Mutually Orthogonal Basis Vector Sets in \(N\)-Dimensions (where \(N \geq 3\)).
3. Calulating Determinant Product of 2 Vectors in 3-Dimensions is same as calculating Cross Product of the 2 Vectors.
4. The Length/Magnitude of the Resultant Vector of a Determinant Product either gives the Area of the Parallelogram whose 2 adjacent sides are given by Vectors in 3-Dimensions or gives the Hyper-Volumes bounded by the Vectors in Higher Dimensions.
5. The Sign of the Resultant Vector of a Determinant Product depends on order in which the Vectors are placed while calculating the Product. Each Change in the Order of Vectors in calculation of Determinant Product results in Changing of the Sign of the Resultant Vector.
6. The following demonstrates calculation of Determinant Product of 3 4-Dimensional Vectors \(\vec{A}\), \(\vec{B}\) and \(\vec{C}\) represented in Identity Orthonormal Basis Vectors \(\hat{e_1}\), \(\hat{e_2}\), \(\hat{e_3}\) and \(\hat{e_4}\) as given below
   
   \(\hat{e_1}=\begin{bmatrix}1\\0\\0\\0\end{bmatrix}\hspace{.5cm}\hat{e_2}=\begin{bmatrix}0\\1\\0\\0\end{bmatrix}\hspace{.5cm}\hat{e_3}=\begin{bmatrix}0\\0\\1\\0\end{bmatrix}\hspace{.5cm}\hat{e_4}=\begin{bmatrix}0\\0\\0\\1\end{bmatrix}\)
   
   \(\vec{A}=A_1\mathbf{\hat{e_1}} + A_2\mathbf{\hat{e_2}} + A_3\mathbf{\hat{e_3}} + A_4\mathbf{\hat{e_4}}\hspace{.5cm}\vec{B}=B_1\mathbf{\hat{e_1}} + B_2\mathbf{\hat{e_2}} + B_3\mathbf{\hat{e_3}} + B_4\mathbf{\hat{e_4}}\hspace{.5cm}\vec{C}=C_1\mathbf{\hat{e_1}} + C_2\mathbf{\hat{e_2}} + C_3\mathbf{\hat{e_3}} + C_4\mathbf{\hat{e_4}}\)
   
   The Determinant Product is calculated by evaluting the following Determinant
   
   \({[\vec{A}\hspace{.2cm}\vec{B}\hspace{.2cm}\vec{C}]}_D=\begin{vmatrix}\mathbf{\hat{e_1}} & \mathbf{\hat{e_2}} & \mathbf{\hat{e_3}} & \mathbf{\hat{e_4}} \\ A_1 & A_2 & A_3 & A_4 \\ B_1 & B_2 & B_3 & B_4 \\ C_1 & C_2 & C_3 & C_4\end{vmatrix}\)
   
   \(\Rightarrow {[\vec{A}\hspace{.2cm}\vec{B}\hspace{.2cm}\vec{C}]}_D=\begin{vmatrix}A_2 & A_3 & A_4 \\B_2 & B_3 & B_4 \\C_2 & C_3 & C_4\end{vmatrix}\mathbf{\hat{e_1}} - \begin{vmatrix}A_1 & A_3 & A_4 \\B_1 & B_3 & B_4 \\C_1 & C_3 & C_4\end{vmatrix}\mathbf{\hat{e_2}} + \begin{vmatrix}A_1 & A_2 & A_4 \\B_1 & B_2 & B_4 \\C_1 & C_2 & C_4\end{vmatrix}\mathbf{\hat{e_3}} - \begin{vmatrix}A_1 & A_2 & A_3 \\B_1 & B_2 & B_3 \\C_1 & C_2 & C_3\end{vmatrix}\mathbf{\hat{e_4}}\)
   
   Since \(\hat{e_1}\), \(\hat{e_2}\), \(\hat{e_3}\) and \(\hat{e_4}\) are Identity Orthonormal Basis Vectors, the following are Determinant Product relation between these Vectors
   
   \(\mathbf{\hat{e_1}}={[\mathbf{\hat{e_2}}\hspace{.2cm}\mathbf{\hat{e_3}}\hspace{.2cm}\mathbf{\hat{e_4}}]}_D\hspace{.5cm} \mathbf{\hat{e_2}}={[\mathbf{\hat{e_1}}\hspace{.2cm}\mathbf{\hat{e_3}}\hspace{.2cm}\mathbf{\hat{e_4}}]}_D\hspace{.5cm} \mathbf{\hat{e_3}}={[\mathbf{\hat{e_1}}\hspace{.2cm}\mathbf{\hat{e_2}}\hspace{.2cm}\mathbf{\hat{e_4}}]}_D\hspace{.5cm} \mathbf{\hat{e_4}}={[\mathbf{\hat{e_1}}\hspace{.2cm}\mathbf{\hat{e_2}}\hspace{.2cm}\mathbf{\hat{e_3}}]}_D\)
   
   The Determinant Product of \(N-1\) Vectors of \(N\)-Dimensions (where \(N \geq 3\)) can be calculated similarly.
7. The following example calculates the Determinant Product of 3 4-Dimensional Vectors \(\vec{A}\), \(\vec{B}\) and \(\vec{C}\) as given below represented in Identity Orthonormal Basis Vectors
   
   \(\vec{A}=3\mathbf{\hat{e_1}} - 2\mathbf{\hat{e_2}} + 1\mathbf{\hat{e_3}} + 3\mathbf{\hat{e_4}}\hspace{.5cm}\vec{B}=-3\mathbf{\hat{e_1}} + 3\mathbf{\hat{e_2}} - 2\mathbf{\hat{e_3}} + 2\mathbf{\hat{e_4}}\hspace{.5cm}\vec{C}=1\mathbf{\hat{e_1}} + 5\mathbf{\hat{e_2}} + 2\mathbf{\hat{e_3}} - 2\mathbf{\hat{e_4}}\)
   
   \({[\vec{A}\hspace{.2cm}\vec{B}\hspace{.2cm}\vec{C}]}_D=\begin{vmatrix}\mathbf{\hat{e_1}} & \mathbf{\hat{e_2}} & \mathbf{\hat{e_3}} & \mathbf{\hat{e_4}} \\ 3 & -2 & 1 & 3 \\ -3 & 3 & -2 & 2 \\ 1 & 5 & 2 & -2\end{vmatrix}\)
   
   \(\Rightarrow {[\vec{A}\hspace{.2cm}\vec{B}\hspace{.2cm}\vec{C}]}_D=\begin{vmatrix}-2 & 1 & 3 \\3 & -2 & 2 \\5 & 2 & -2\end{vmatrix}\mathbf{\hat{e_1}} - \begin{vmatrix}3 & 1 & 3 \\-3 & -2 & 2 \\1 & 2 & -2\end{vmatrix}\mathbf{\hat{e_2}} + \begin{vmatrix}3 & -2 & 3 \\-3 & 3 & 2 \\1 & 5 & -2\end{vmatrix}\mathbf{\hat{e_3}} - \begin{vmatrix}3 & -2 & 1 \\-3 & 3 & -2 \\1 & 5 & 2\end{vmatrix}\mathbf{\hat{e_4}}\)
   
   \(\Rightarrow {[\vec{A}\hspace{.2cm}\vec{B}\hspace{.2cm}\vec{C}]}_D=64\mathbf{\hat{e_1}} + 16\mathbf{\hat{e_2}} - 94\mathbf{\hat{e_3}} - 22\mathbf{\hat{e_4}}\)