The vector triple product (VTP) is defined as a x (b x c).

Expansion (BAC-CAB rule): a x (b x c) = (a.c)b - (a.b)c.

Key observations:

• The result is a linear combination of b and c — it lies in the plane of b and c.
• The result is perpendicular to a (since it's a cross product with a on the left).
• Actually, a x (b x c) is perpendicular to a AND perpendicular to b x c. But since b x c is perpendicular to the plane of b,c, and the result lies IN the plane of b,c, both conditions are consistent.

Caution: The cross product is NOT associative. (a x b) x c = (a.c)b - (b.c)a — this lies in the plane of a and b, not b and c.

Mnemonic: "The outer vector dots with the far one first, multiplied by the middle vector, minus the outer dotted with the middle, multiplied by the far vector."