If a⃗ and b⃗ makes an angle cos⁻¹(5/9) with each other, then |a⃗+b⃗|=√2|a⃗-b⃗| for |a⃗|=n|b⃗| The integer value of n is _________.

Solution

Physics sir · 4 steps

Square both sides to turn the magnitudes into dot products, then put |a⃗| = n|b⃗|.

1. Square both sides. Use |a⃗ ± b⃗|² = a² + b² ± 2ab cos θ. Squaring turns the √2 on the right into a factor of 2.

   a² + b² + 2ab cos θ = 2(a² + b² - 2ab cos θ)

2. Rearrange. This gives a² + b² = 6ab cos θ = 6ab × 5/9 = 10/3ab.
3. Use the ratio. Put a = nb and divide by b² to get n² + 1 = 10/3n, that is 3n² - 10n + 3 = 0. This factorises as (3n - 1)(n - 3) = 0, so n = 3 or n = 1/3.
4. So. Only one root is an integer, so n = 3.

The idea to keep. Whenever you see a relation between |a⃗+b⃗| and |a⃗-b⃗|, square it first. You get a quadratic in the ratio of the magnitudes.

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