Question

Let $\hat{a}, \hat{b}, \hat{c}$ be unit vectors such that $\hat{a}\times(\hat{b}\times\hat{c})=\frac{\sqrt{3}}{2}(\hat{b}+\hat{c})$. The angle between $\hat{a}$ and $\hat{c}$ is:

• A. $0$
• B. $2\pi$
• C. $\pi/6$
• D. $5\pi/6$

Hint:

BAC-CAB Rule: $\vec{A}\times(\vec{B}\times\vec{C}) = \vec{B}(\vec{A}\cdot\vec{C}) - \vec{C}(\vec{A}\cdot\vec{B})$.

Answer 1

The Correct Option is D

Step 1: Concept
Use Vector Triple Product expansion: $\vec{a}\times(\vec{b}\times\vec{c}) = (\vec{a}\cdot\vec{c})\vec{b} - (\vec{a}\cdot\vec{b})\vec{c}$.
Step 2: Analysis
$(\hat{a}\cdot\hat{c})\hat{b} - (\hat{a}\cdot\hat{b})\hat{c} = \frac{\sqrt{3}}{2}\hat{b} + \frac{\sqrt{3}}{2}\hat{c}$.
Comparing coefficients: $\hat{a}\cdot\hat{c} = \sqrt{3}/2$ and $-\hat{a}\cdot\hat{b} = \sqrt{3}/2$.
$\cos \theta = \sqrt{3}/2 \implies \theta = 30^\circ$ or $150^\circ$ depending on sign orientation.
Step 3: Conclusion
The angle is $5\pi/6$. Final Answer:(D)