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Scalar Quad Product

1. The Scalar Quad Product of 4 3-Dimensional Vectors \(\vec{A}\), \(\vec{B}\), \(\vec{C}\) and \(\vec{D}\) is calculated as follows
   
   \((\vec{A} \times \vec{B}) \cdot (\vec{C} \times \vec{D})=\begin{vmatrix} \vec{A}\cdot\vec{C} & \vec{A}\cdot\vec{D} \\\vec{B}\cdot\vec{C} & \vec{B}\cdot\vec{D}\end{vmatrix} =(\vec{A}\cdot\vec{C})(\vec{B}\cdot\vec{D})-(\vec{A}\cdot\vec{D})(\vec{B}\cdot\vec{C})\)
2. The Scalar Quad Product can be used to calculate Square of the Magnitude of the Cross Product Between 2 Vectors \(\vec{A}\) and \(\vec{B}\) as follows
   
   \(|\vec{A} \times \vec{B}|^2=(\vec{A} \times \vec{B}) \cdot (\vec{A} \times \vec{B})=(\vec{A}\cdot\vec{A})(\vec{B}\cdot\vec{B})-{(\vec{A}\cdot\vec{B})}^2\)
   
   Please note that this is same as calculating the Determinant of Metric Tensor of a Matrix containing Vectors \(\vec{A}\) and \(\vec{B}\) as it's Columns.