Let a⃗ = a₁î + a₂ĵ + a₃k̂ aᵢ > 0, i = 1,2,3 be a vector which makes equal angles with the coordinate axes OX, OY and OZ. Also, let the projection of a⃗ on the vector 3î + 4ĵ be 7. Let b⃗ be a vector obtained by rotating a⃗ with 90°. If a⃗, b⃗ and x-axis are coplanar, then projection of a vector b⃗ on 3î + 4ĵ is equal to:

Solution

Maths sir · 4 steps

Equal angles means a⃗ = λ(1,1,1). Coplanar with the x-axis means b⃗ = μa⃗ + νî.

1. Find a. Equal angles with all three axes and positive components give a⃗ = λ(î + ĵ + k̂). Its projection on 3î + 4ĵ is (3λ + 4λ)/5 = 7λ/5 = 7. So λ = 5, a⃗ = 5î + 5ĵ + 5k̂ and |a⃗| = 5√3.
2. Shape of b. b⃗ lies in the plane of a⃗ and î, so b⃗ = μa⃗ + νî = (5μ + ν)î + 5μĵ + 5μk̂. A rotation by 90° makes b⃗ ⊥ a⃗ and keeps the length, so |b⃗| = 5√3.

   a⃗·b⃗ = 0 ⇒ ν = -15μ ⇒ b⃗ = -10μ î + 5μ ĵ + 5μ k̂

3. Fix the length. |b⃗|² = 150μ² = 75, so μ = ±1/√2. Then the projection of b⃗ on 3î + 4ĵ is (-30μ + 20μ)/5 = -2μ = ∓√2.
4. So. Rotating in the sense that makes the projection positive, with μ = -1/√2, gives b⃗ = 5/√2(2î - ĵ - k̂). Its projection on 3î + 4ĵ is √2, which is option (b).

The idea to keep. A vector coplanar with p⃗ and q⃗ can be written as μp⃗ + νq⃗. Rotation by 90° gives you two conditions: dot product zero and the same length.

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