Let a⃗ = î - 2ĵ + k̂ and b⃗ = î - ĵ + k̂ be two vectors. If c⃗ is a vector such that b⃗ × c⃗ = b⃗ × a⃗ and c⃗ .a⃗ = 0, then c⃗ .b⃗ is equal to

Solution

Maths sir · 4 steps

Equal cross products with b⃗ make c⃗ - a⃗ parallel to b⃗. Start from there.

1. Parallel difference. Move everything to one side: b⃗ × (c⃗ - a⃗) = 0⃗. A zero cross product means c⃗ - a⃗ is parallel to b⃗, so c⃗ = a⃗ + λ b⃗ for some scalar λ.
2. Needed dot products. |a⃗|² = 1 + 4 + 1 = 6, a⃗ · b⃗ = 1 + 2 + 1 = 4 and |b⃗|² = 1 + 1 + 1 = 3.

   c⃗ · a⃗ = |a⃗|² + λ (a⃗ · b⃗) = 6 + 4λ = 0 ⇒ λ = -3/2

3. Dot with b. c⃗ · b⃗ = a⃗ · b⃗ + λ |b⃗|² = 4 + (-3/2)(3) = 4 - 9/2.
4. So. c⃗ · b⃗ = -1/2, which is option (a).

The idea to keep. Whenever b⃗ × c⃗ = b⃗ × a⃗, do not cancel b⃗. Write c⃗ = a⃗ + λ b⃗ and let the other condition fix λ.

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