If a, b, c are three non-coplanar vectors, then (a+b+c) . [(a+b) × (a+c)] equals:

If a, b, c are three non-coplanar vectors, then (a+b+c) . [(a+b) × (a+c)] equals:

• A. −[a b c]
• B. [a b c]
• C. 2[a b c]
• D. 0

Answer: A) −[a b c]

Explanation: Expand the cross product: (a+b) × (a+c) = a×a + a×c + b×a + b×c = 0 + a×c + b×a + b×c. Dot product with (a+b+c): The only non-zero scalar triple products surviving will be c.(b×a) and b.(a×c). c.(b×a) = [c b a] = −[a b c]. b.(a×c) = [b a c] = −[a b c].