Let a⃗=3 î-ĵ+2 k̂, b⃗=a⃗ ×(î-2 k̂) and c⃗=b⃗ × k̂. Then the projection of c⃗-2 ĵ on a⃗ is :

Solution

Maths sir · 3 steps

Build b⃗ and c⃗ one cross product at a time, then dot with a⃗ and divide by |a⃗|.

1. Find b. With a⃗ = (3, -1, 2) and î - 2k̂ = (1, 0, -2), the cross product gives b⃗ = ((-1)(-2) - 2 × 0, 2 × 1 - 3 × (-2), 3 × 0 - (-1) × 1) = (2, 8, 1). Check: b⃗ · a⃗ = 6 - 8 + 2 = 0, as it must be.
2. Find c. c⃗ = b⃗ × k̂ = (2, 8, 1) × (0, 0, 1) = (8 × 1 - 1 × 0, 1 × 0 - 2 × 1, 0) = (8, -2, 0). So c⃗ - 2ĵ = (8, -4, 0).

   Projection = ((c⃗ - 2ĵ) · a⃗)/|a⃗| = (24 + 4 + 0)/√(9 + 1 + 4) = 28/√14

3. So. 28/√14 = 28√14/14 = 2√14, which is option (d).

The idea to keep. Projection of p⃗ on a⃗ is (p⃗ · a⃗)/|a⃗|. After every cross product, check it is perpendicular to both factors; a zero dot product catches sign slips early.

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