Question

If \( \vec{a} \) and \( \vec{b} \) are two unit vectors and \( \theta \) is the angle between them, then \( |\vec{a} + \vec{b}| \) is a unit vector if \( \theta \) is:

• A. $\pi/3$
• B. $\pi/2$
• C. $2\pi/3$
• D. $\pi/4$

Hint:

If the sum of two unit vectors is also a unit vector, they must be at $120^{\circ}$ to each other.

Answer 1

The Correct Option is C

Step 1: Concept
$|\vec{a} + \vec{b}|^{2} = |\vec{a}|^{2} + |\vec{b}|^{2} + 2|\vec{a}||\vec{b}|\cos\theta$.
Step 2: Analysis
Since all are unit vectors, $1^{2} = 1^{2} + 1^{2} + 2(1)(1)\cos\theta$. $1 = 2 + 2\cos\theta \Rightarrow -1 = 2\cos\theta \Rightarrow \cos\theta = -1/2$.
Step 3: Conclusion
$\theta = \cos^{-1}(-1/2) = 120^{\circ}$ or $2\pi/3$.
Final Answer: (C)