Vector Quad Product

1. The Vector Quad Product of 4 3-Dimensional Vectors \(\vec{A}\), \(\vec{B}\), \(\vec{C}\) and \(\vec{D}\) is calculated as follows
   
   \((\vec{A} \times \vec{B}) \times (\vec{C} \times \vec{D})=((\vec{A} \times \vec{B})\cdot\vec{D})\vec{C} - ((\vec{A} \times \vec{B})\cdot\vec{C})\vec{D} =[\vec{A}\hspace{.1cm}\vec{B}\hspace{.1cm}\vec{D}]\hspace{.1cm}\vec{C}-[\vec{A}\hspace{.1cm}\vec{B}\hspace{.1cm}\vec{C}]\hspace{.1cm}\vec{D}\)
2. Following gives the Derivation of the Formula for Vector Quad Product
   
   Let \(\vec{P}= \vec{A} \times \vec{B}\)    ...(1)
   
   \(\Rightarrow(\vec{A} \times \vec{B}) \times (\vec{C} \times \vec{D})=\vec{P} \times (\vec{C} \times \vec{D})\)
   
   \(\Rightarrow(\vec{A} \times \vec{B}) \times (\vec{C} \times \vec{D})=(\vec{P} \cdot\vec{D})\vec{C} - (\vec{P} \cdot\vec{C})\vec{D})\)    ...(2)
   
   Substituting the value of \(\vec{P}\) from equation (1) in equation (2) we get
   
   \((\vec{A} \times \vec{B}) \times (\vec{C} \times \vec{D})=((\vec{A} \times \vec{B})\cdot\vec{D})\vec{C} - ((\vec{A} \times \vec{B})\cdot\vec{C})\vec{D} =[\vec{A}\hspace{.1cm}\vec{B}\hspace{.1cm}\vec{D}]\hspace{.1cm}\vec{C}-[\vec{A}\hspace{.1cm}\vec{B}\hspace{.1cm}\vec{C}]\hspace{.1cm}\vec{D}\)