Question

Let $\vec a \ and \ \vec b$ be two unit vectors and θ is the angle between them.Then $\vec a + \vec b$ is a unit vector if
(A)θ= $\frac{\pi}{4}$
(B)θ= $\frac{\pi}{3}$
(C)θ= $\frac{\pi}{2}$
(D)θ= $\frac{2\pi}{3}$

Answer 1

Given,

a and b are unit vectors, then |a| = 1 and |b| = 1.

Now,

|a + b|² = |a|²  + |b|² + 2|a||b|Cosθ

∴ |a + b|² = 1 + 1 + 2Cosθ

∴ 1 = 2 + 2Cosθ

∴ Cosθ = -1/2

Cosθ = Cos(120)

∴ θ = 120 or 2π/3

Hence, Correct answer is Option (D).

Hope it helps.