(b) Prove \begin{array}{l}{\vec{(a} \times \vec{b} ) \cdot(\vec{c} \ti

1.	(b) Prove \begin{array}{l}{\vec{(a} \times \vec{b} ) \cdot(\vec{c} \times \vec{d})} \\ {\vec{a} \cdot \vec{c} )(\vec{b} \cdot \vec{d})-(\vec{a} \cdot \vec{d})(\vec{b} \cdot \vec{c})}\end{array}
Answer» We need to prove that (A x B).(C x D) = (A.C)(B.D) - (A.D)(B.C) Lets consider (A x B) = Pnow RHS = P.(C x D) = (P x C).D ={(A x B)x C}.D ={B(A.C) - A(B.C)}.D [vector triple product expansion] =(B.D)(A.C) - (A.D)(B.C) = LHS

(b) Prove \begin{array}{l}{\vec{(a} \times \vec{b} ) \cdot(\vec{c} \times \vec{d})} \\ {\vec{a} \cdot \vec{c} )(\vec{b} \cdot \vec{d})-(\vec{a} \cdot \vec{d})(\vec{b} \cdot \vec{c})}\end{array}

Answer»

We need to prove that (A x B).(C x D) = (A.C)(B.D) - (A.D)(B.C)

Lets consider (A x B) = Pnow RHS = P.(C x D)

= (P x C).D

={(A x B)x C}.D

={B(A.C) - A(B.C)}.D [vector triple product expansion]

=(B.D)(A.C) - (A.D)(B.C)

= LHS