The relevant calculations are as follows.

$\mathbf{A}\times \mathbf{B}\= \|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ 3& -2& 4\\ 5& 7& -6\end{array}\|$= $\left[\begin{array}{c}-16\\ 38\\ 31\end{array}\right]$ and $\mathbf{C}\times \mathbf{V}\= \|\left[\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ 4& -3& -5\\ 2& -5& 3\end{array}\right]\|$ = $\left[\begin{array}{c}-34\\ -22\\ -14\end{array}\right]$ 

$\left(\mathbf{A}\times \mathbf{B}\right)·\left(\mathbf{C}\mathbf{\times }\mathbf{V}\right)$= $16\cdot 34-38\cdot 22-31\cdot 14\= -726$ 

$\begin{array}{c}\mathbf{A}·\mathbf{C}\=12\+6-20\=-2\\ \mathbf{A}·\mathbf{V}\=6\+10\+12\=28\\ \mathbf{B}·\mathbf{C}\=20-21\+30\=29\\ \mathbf{B}·\mathbf{V}\=10-35-18\=-43\end{array}\}$ ⇒ $\left|\begin{array}{cc}\mathbf{A}·\mathbf{C}& \mathbf{A}·\mathbf{V}\\ \mathbf{B}·\mathbf{C}& \mathbf{B}·\mathbf{V}\end{array}\right|$ = $\|\begin{array}{cc}-2& 28\\ 29& -43\end{array}\|$ = $-726$ 

It would be more natural to state the identity in terms of A, B, C, and D, but the symbol "D" is reserved in Maple for the differentiation operator. There are ways around this restriction, but the cure is worse than the illness.

To write the notation for the determinant, place single strokes around the matrix, fill in the entries of the matrix, then delete the brackets that denote "matrix". It is not essential to remove these brackets - Maple will still interpret the single vertical strokes as a call to the determinant function. But it looks nicer to delete the brackets.

Apply the DotProduct, CrossProduct, Determinant, and Matrix commands. The syntax used below in the Matrix command provides each row as a list-of-lists, with the sublists defining the rows of the matrix.

It would be more natural to state the identity in terms of A, B, C, and D, but the symbol "D" is reserved in Maple for the differentiation operator. There are ways around this restriction, but the cure is worse than the illness.