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.

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=(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!

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

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