The resultant of two vectors A⃗ and B⃗ is perpendicular to A⃗ and its magnitude is half that of B⃗. The angle between vectors A⃗ and B⃗ is _________°.

Solution

Physics sir · 3 steps

Turn both conditions into dot-product equations, and the angle drops out of the cosine.

1. Perpendicular condition. Let R⃗ = A⃗ + B⃗ and let θ be the angle between A⃗ and B⃗. Since R⃗ ⊥ A⃗, R⃗ · A⃗ = 0, so A² + AB cos θ = 0, which gives AB cos θ = -A².
2. Magnitude condition. Square the resultant: R² = A² + B² + 2AB cos θ = A² + B² - 2A² = B² - A². With R = B/2, this gives B²/4 = B² - A², so A = √3/2B.

   cos θ = -A/B = -√3/2

3. So. The cosine is negative, so θ is obtuse: θ = 180° - 30° = 150°.

The common slip: 30. Dropping the minus sign and taking cos θ = √3/2 gives 30, which is the angle inside the right triangle, not the angle between the vectors.

The idea to keep. Perpendicular means the dot product is zero, and magnitude conditions mean you square the resultant. Then keep the sign of cos θ: a negative cosine always gives an obtuse angle.

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