Consider two vectors u⃗=3 î-ĵ and v⃗=2 î+ĵ-λ k̂, λ>0. The angle between them is given by cos ⁻¹(√5/(2 √7)). Let v⃗=v⃗₁+v₂⃗, where v⃗₁ is parallel to u⃗ and v₂⃗ is perpendicular to u⃗. Then the value |v₁⃗|²+|v₂⃗|² is equal to

Solution

Maths sir · 3 steps

The pieces are perpendicular, so their squared lengths add to |v⃗|². Find λ first.

1. Pythagoras for vectors. Since v₁⃗ ⊥ v₂⃗, we have v₁⃗·v₂⃗ = 0. Then |v⃗|² = |v₁⃗+v₂⃗|² = |v₁⃗|² + |v₂⃗|². So you only need |v⃗|² = 5 + λ².
2. Use the angle. u⃗·v⃗ = 6 - 1 + 0 = 5, |u⃗| = √10 and |v⃗| = √(5+λ²). Set cos θ equal to the given value and square both sides.

   25/10(5+λ²) = 5/28 ⇒ 50(5+λ²) = 700 ⇒ λ² = 9

3. So. |v₁⃗|²+|v₂⃗|² = |v⃗|² = 5 + 9 = 14, which is option (d).

The idea to keep. When a vector is split into a part along u⃗ and a part perpendicular to u⃗, the squared lengths of the parts add up to |v⃗|². You do not need to find the parts themselves.

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