If two vectors A⃗ and B⃗ having equal magnitude R are inclined at angle θ, then

Solution

Physics sir · 4 steps

With equal magnitudes, the sum formula folds into a half-angle cosine. Check it with θ = 0.

1. Magnitude of the sum. For two vectors at angle θ, |A⃗+B⃗|² = A² + B² + 2AB cos θ. With A = B = R this becomes 2R²(1+cos θ).

   |A⃗+B⃗| = √2R²(1+cos θ) = √(4R²cos² θ/2) = 2R cos θ/2

2. Magnitude of the difference. For the difference the sign of the cosine term flips: |A⃗-B⃗|² = 2R²(1-cos θ) = 4R²sin² θ/2. So |A⃗-B⃗| = 2R sin θ/2, which matches neither (b) nor (c).
3. Quick check. At θ = 0 the vectors point the same way. The sum must be 2R and the difference must be 0. Only 2R cos θ/2 for the sum and 2R sin θ/2 for the difference pass this test.
4. So. The correct relation is |A⃗+B⃗| = 2R cos θ/2, option (a).

The idea to keep. For equal vectors, the sum is 2R cos θ/2 and the difference is 2R sin θ/2. If you forget which is which, put θ = 0 and see which one gives zero.

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