If x⃗ and y⃗ be two non-zero vectors such that | x⃗ + y⃗ | = | x⃗ | and 2x⃗ + λ y⃗ is perpendicular to y⃗ , then the value of λ is _________ .

Solution

Maths sir · 3 steps

Square the modulus condition: it hands you x⃗ · y⃗ in terms of |y⃗|², and λ follows.

1. Square the modulus. Squaring |x⃗ + y⃗| = |x⃗| gives |x⃗|² + 2 x⃗ · y⃗ + |y⃗|² = |x⃗|². The |x⃗|² cancels, so 2 x⃗ · y⃗ = -|y⃗|².
2. Use perpendicularity. Perpendicular vectors have zero dot product, so (2x⃗ + λ y⃗) · y⃗ = 0.

   2 x⃗ · y⃗ + λ |y⃗|² = 0 ⇒ -|y⃗|² + λ |y⃗|² = 0

3. So. Since y⃗ is non-zero, |y⃗|² ≠ 0, and you can divide by it to get λ - 1 = 0. So λ = 1.

The common slip: 2. Expanding |x⃗ + y⃗|² with only x⃗ · y⃗ in the middle, instead of 2 x⃗ · y⃗, gives λ = 2.

The idea to keep. Whenever you see a condition on |a⃗ + b⃗|, square it at once. That turns it into a dot product you can substitute into the next condition.

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