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:

Let A, B and C be vectors in R3.


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=(Bx, 0, 0), C=(Cx, Cy, 0), A=(Ax, Ay, Az)

B×C=(0, 0, BxCy)

A×(B×C)=(Ax, Ay, Az)×(0, 0, BxCy)
               =(AyBxCy, -AxBxCy, 0)

B(A·C)-C(A·B)=[(AxCx+AyCy)Bx, 0, 0] - [AxBxCx, AxBxCy, 0]
                         =(AyBxCy, -AxBxCy, 0)

A×(B×C)=B(A·C)-C(A·B) QED

A few tidbits:

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


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.

Bressoud, D. Second Year Calculus © 1991