Vector triple product (idea)

The vector triple product, also known as the BAC-CAB identity, is used to calculate the cross product when three vectors are involved. First, a few words about notation:

• All vectors are in bold, all scalars are not.
• A·B is the dot product of A with B.
• A×B is the cross product of A with B.

Let A, B and C be vectors in R³.

A×(B×C)=B(A·C)-C(A·B)

This is much easier to prove if we pick our coordinate system such that B lies along the x-axis and C lies in the x-y plane. Since we are dealing with proper vectors, if we show this is true in one coordinate system, it must be true in all coordinate systems.

B=(B_x, 0, 0), C=(C_x, C_y, 0), A=(A_x, A_y, A_z)

B×C=(0, 0, B_xC_y)

A×(B×C)=(A_x, A_y, A_z)×(0, 0, B_xC_y)
               =(A_yB_xC_y, -A_xB_xC_y, 0)

B(A·C)-C(A·B)=[(A_xC_x+A_yC_y)B_x, 0, 0] - [A_xB_xC_x, A_xB_xC_y, 0]
                         =(A_yB_xC_y, -A_xB_xC_y, 0)

A×(B×C)=B(A·C)-C(A·B) QED

A few tidbits:

(A×B)×C= -C×(A×B)
               = B(A·C)- A(B·C)       Parentheses are important!

A×(B×C)+B×(C×A)+C×(A×B)=0

All higher vector products (products involving more than three vectors) can be reduced to expressions containing no more than one cross product per term, often due to the BAC-CAB identity.

Sources:
Bressoud, D. Second Year Calculus © 1991
http://farside.ph.utexas.edu/teaching/em1/lectures/node11.html