Question

If $\bar{a}$ and $\bar{b}$ are unit vectors and $\theta$ is the angle between them, then $\bar{a} + \bar{b}$ is a unit vector when $\theta$ is

• A. $\frac{\pi}{3}$
• B. $\frac{2\pi}{3}$
• C. $\frac{\pi}{2}$
• D. $\frac{\pi}{4}$

Hint:

If two unit vectors form an equilateral triangle with their sum, the angle between the vectors must be $120^\circ$.

Answer 1

The Correct Option is B

Step 1: Concept
The magnitude of the sum of two vectors is given by $|\bar{a} + \bar{b}|^2 = |\bar{a}|^2 + |\bar{b}|^2 + 2|\bar{a}||\bar{b}|\cos\theta$.

Step 2: Meaning
Since $\bar{a}$, $\bar{b}$, and $\bar{a} + \bar{b}$ are all unit vectors, their magnitudes are all equal to 1.

Step 3: Analysis
Substitute the magnitudes into the formula: $1^2 = 1^2 + 1^2 + 2(1)(1)\cos\theta$. $1 = 2 + 2\cos\theta$. $-1 = 2\cos\theta \implies \cos\theta = -1/2$.

Step 4: Conclusion
$\theta = \cos^{-1}(-1/2) = 120^\circ = \frac{2\pi}{3}$. Final Answer: (B)