JEE Main 2022 · 25 June · morning · Single correct · +4 / −1
Which of the following relations is true for two unit vector A and B making an angle θ to each other?
- (a) |A+B|=|A-B|tan θ/2
- (b) |A-B|=|A+B|tan θ/2
- (c) |A+B|=|A-B|cosθ/2
- (d) |A-B|=|A+B|cos θ/2
Answer and solution
Answer (b) |A-B|=|A+B|tan θ/2
Solution
Physics sir · 3 steps
Find both magnitudes in half-angle form; their ratio turns straight into tan θ/2.
- Magnitude of the sum. For unit vectors, |A+B|² = 1 + 1 + 2 cos θ = 2(1+cos θ) = 4 cos² θ/2. So |A+B| = 2 cos θ/2.
- Magnitude of the difference. Similarly, |A-B|² = 1 + 1 - 2 cos θ = 2(1-cos θ) = 4 sin² θ/2. So |A-B| = 2 sin θ/2.
(|A-B|)/(|A+B|) = (2 sin θ/2)/(2 cos θ/2) = tan θ/2
- So. |A-B| = |A+B|tan θ/2, which is option (b).
The idea to keep. For two unit vectors at angle θ, remember |A+B| = 2 cos θ/2 and |A-B| = 2 sin θ/2. Any relation between them follows from these two.
Answer from the NTA final answer key. Solution by PrepWiser.