Let a⃗ = - î - ĵ + k̂,a⃗ . b⃗ = 1 and a⃗ × b⃗ = î - ĵ. Then a⃗ - 6b⃗ is equal to :

Solution

Maths sir · 4 steps

Cross a⃗ with a⃗ × b⃗ and expand; that pulls b⃗ out with no unknown components.

1. Use triple product. Expand a⃗ × (a⃗ × b⃗). It brings in a⃗ · b⃗ and |a⃗|², both of which you already know. Here a⃗ · b⃗ = 1 and |a⃗|² = 1 + 1 + 1 = 3.

   a⃗ × (a⃗ × b⃗) = (a⃗ · b⃗) a⃗ - |a⃗|² b⃗ = a⃗ - 3b⃗

2. Compute left side. (-î - ĵ + k̂) × (î - ĵ) = î + ĵ + 2k̂. So 3b⃗ = a⃗ - (î + ĵ + 2k̂) = -2î - 2ĵ - k̂.
3. Find the vector. a⃗ - 6b⃗ = a⃗ - 2(3b⃗) = (-î - ĵ + k̂) + (4î + 4ĵ + 2k̂) = 3î + 3ĵ + 3k̂. Check: b⃗ = -1/3(2î + 2ĵ + k̂) gives a⃗ · b⃗ = 2/3 + 2/3 - 1/3 = 1.
4. So. a⃗ - 6b⃗ = 3(î + ĵ + k̂), which is option (a).

The idea to keep. When you know both a⃗ · b⃗ and a⃗ × b⃗, cross a⃗ with the cross product and expand. b⃗ comes out directly, with no three unknowns to solve for.

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