Let a⃗ and b⃗ be two vectors, Let |a⃗|=1,|b⃗|=4 and a⃗ · b⃗=2. If c⃗=(2 a⃗ × b⃗)-3 b⃗, then the value of b⃗ · c⃗ is :

Solution

Maths sir · 3 steps

Dot b⃗ into each piece: the cross-product term dies at once, so only -3|b⃗|² survives.

1. Split the product. The dot product distributes over addition, so b⃗ · c⃗ = 2 b⃗ · (a⃗ × b⃗) - 3 b⃗ · b⃗.
2. Cross term vanishes. a⃗ × b⃗ is perpendicular to b⃗, so b⃗ · (a⃗ × b⃗) = 0. It is a scalar triple product with b⃗ repeated.

   b⃗ · c⃗ = 2(0) - 3|b⃗|² = -3 × 16

3. So. With |b⃗| = 4, |b⃗|² = 16 and b⃗ · c⃗ = -48, which is option (a). The values |a⃗| = 1 and a⃗ · b⃗ = 2 are never needed.

The idea to keep. A vector dotted with a cross product that already contains it is always zero. Spot that first and a long-looking problem shrinks to one line.

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