Vector A Dot Vector B) Dot (Vector A CROSS Vector B)

Let us resolve A and B vectors into 3 components i , j and k

Hence, Vector A = A_x i + A_y j + A_z k

Vector B = B_x i + B_y j + B_z k

As (a) i Dot i = i (b)  i Dot j = 0 (c) i Dot k = 0

Similarly, (a) i X i = 0 (b)  i X  j = k (c) i X k = -j (d) j X i = -k (e) k X i = j

A Dot B = A_xB_x i + A_yB_y j + A_zB_z k

A X B = A_xB_y k + A_xB_z j – A_yB_x k + A_yB_z i - A_xB_x j – A_zB_y i

(Vector A Dot Vector B) Dot (Vector A CROSS Vector B) = (A_xB_x i + A_yB_y j + A_zB_z k) Dot (A_xB_y k + A_xB_z j – A_yB_x k + A_yB_z i - A_xB_x j – A_zB_y i )

                       = A_xB_x(A_yB_z  - A_zB_y) i + A_yB_y(A_xB_z  - A_zB_x) j + A_zB_z(A_xB_y  - A_yB_x) k