For λ>0, let θ be the angle between the vectors a⃗=î+λ ĵ-3 k̂ and b⃗=3 î-ĵ+2 k̂. If the vectors a⃗+b⃗ and a⃗-b⃗ are mutually perpendicular, then the value of (14 cos θ)² is equal to

Solution

Maths sir · 4 steps

Perpendicular sum and difference means a⃗ and b⃗ have equal lengths. Use that to find λ.

1. Perpendicular condition. Perpendicular vectors have zero dot product. Expanding, (a⃗+b⃗)·(a⃗-b⃗) = |a⃗|² - |b⃗|², because the cross terms a⃗·b⃗ and b⃗·a⃗ cancel. So |a⃗| = |b⃗|.
2. Find lambda. |a⃗|² = 1 + λ² + 9 = 10 + λ² and |b⃗|² = 9 + 1 + 4 = 14. Setting them equal gives λ² = 4, and since λ > 0, λ = 2.
3. Dot product. a⃗·b⃗ = (1)(3) + (2)(-1) + (-3)(2) = 3 - 2 - 6 = -5.

   cos θ = (a⃗·b⃗)/(|a⃗| |b⃗|) = -5/(√14×√14) = -5/14

4. So. 14 cos θ = -5, so (14 cos θ)² = 25. The answer is (a).

The idea to keep. Whenever a⃗+b⃗ is perpendicular to a⃗-b⃗, read it straight away as |a⃗| = |b⃗|. The geometry is a rhombus whose diagonals are perpendicular.

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