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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.