Question

If $\vec{a}$, $\vec{b}$ and $(\vec{a} + \vec{b})$ are all unit vectors and $\theta$ is the angle between $\vec{a}$ and $\vec{b}$, then the value of $\theta$ is:

• A. $\frac{2\pi}{3}$
• B. $\frac{5\pi}{6}$
• C. $\frac{\pi}{3}$
• D. $\frac{\pi}{6}$

Hint:

Always use $| \vec{a} + \vec{b} |^2$ and expand using the dot product formula.

Answer 1

The Correct Option is A

Given that $\vec{a}$ and $\vec{b}$ are unit vectors: \[ |\vec{a}| = |\vec{b}| = 1, | \vec{a} + \vec{b} | = 1. \] Now, \[ | \vec{a} + \vec{b} |^2 = (\vec{a} + \vec{b}) \cdot (\vec{a} + \vec{b}) = |\vec{a}|^2 + |\vec{b}|^2 + 2\, \vec{a} \cdot \vec{b}. \] So, \[ 1 = 1 + 1 + 2 (\cos\theta) \implies 1 = 2 + 2\cos\theta. \] Therefore, \[ 2\cos\theta = -1 \implies \cos\theta = -\frac{1}{2}. \] So, \[ \theta = \cos^{-1}\Big(-\frac{1}{2}\Big) = \frac{2\pi}{3}. \]