Let a⃗ = î + ĵ + k̂ and b⃗ = ĵ - k̂. If c⃗ is a vector such that a⃗ × c⃗ = b⃗ and a⃗ .c⃗ = 3, then a⃗ .(b⃗ × c⃗ ) is equal to :

Solution

Maths sir · 4 steps

Rotate the box product so that a⃗ × c⃗ appears; then it is just b⃗ dotted with itself.

1. Cyclic shift. A scalar triple product does not change when you rotate its three vectors cyclically. So a⃗ · (b⃗ × c⃗) = b⃗ · (c⃗ × a⃗).
2. Use the given cross. Swapping the order of a cross product flips its sign, so c⃗ × a⃗ = -(a⃗ × c⃗) = -b⃗.

   a⃗ · (b⃗ × c⃗) = b⃗ · (-b⃗) = -|b⃗|² = -(0² + 1² + (-1)²) = -2

3. Quick check. Solving fully gives c⃗ = 1/3(5î + 2ĵ + 2k̂), which satisfies a⃗ · c⃗ = 3. Then b⃗ × c⃗ = 1/3(4î - 5ĵ - 5k̂), and dotting with a⃗ gives (4 - 5 - 5)/3 = -2.
4. So. a⃗ · (b⃗ × c⃗) = -2, which is option (a). The condition a⃗ · c⃗ = 3 was not even needed.

The idea to keep. Before solving for an unknown vector, try rotating the box product [a⃗ b⃗ c⃗] until a given cross product shows up. Watch the sign every time you swap two vectors.

Answer from the NTA final answer key. Solution by PrepWiser.