Question

If \(\vec{a},\vec{b},\vec{c}\) be three unit vectors such that \(\vec{a}\times(\vec{b}\times\vec{c})=\frac{1}{2}\vec{b}\), \(\vec{b}\) and \(\vec{c}\) being non-parallel. If \(\theta_1\) is the angle between \(\vec{a}\) and \(\vec{b}\) and \(\theta_2\) is the angle between \(\vec{a}\) and \(\vec{c}\), then

• A. \(\theta_1=\frac{\pi}{6},\theta_2=\frac{\pi}{3}\)
• B. \(\theta_1=\frac{\pi}{3},\theta_2=\frac{\pi}{6}\)
• C. \(\theta_1=\frac{\pi}{2},\theta_2=\frac{\pi}{3}\)
• D. \(\theta_1=\frac{\pi}{2},\theta_2=\frac{\pi}{2}\)

Hint:

If \(\vec{b}\) and \(\vec{c}\) are non-parallel, compare coefficients in \(\vec{b}(\vec{a}\cdot\vec{c})-\vec{c}(\vec{a}\cdot\vec{b})\).

Answer 1

The Correct Option is C

Step 1: Use vector triple product identity.
\[ \vec{a}\times(\vec{b}\times\vec{c})=\vec{b}(\vec{a}\cdot\vec{c})-\vec{c}(\vec{a}\cdot\vec{b}) \]
Step 2: Given condition.
\[ \vec{b}(\vec{a}\cdot\vec{c})-\vec{c}(\vec{a}\cdot\vec{b})=\frac{1}{2}\vec{b} \]
Step 3: Compare coefficients of \(\vec{b}\) and \(\vec{c}\).
Since \(\vec{b}\) and \(\vec{c}\) are non-parallel, they are linearly independent.
So coefficient of \(\vec{c}\) must be zero:
\[ -(\vec{a}\cdot\vec{b})=0 \Rightarrow \vec{a}\cdot\vec{b}=0 \]
\[ \Rightarrow \cos\theta_1=0 \Rightarrow \theta_1=\frac{\pi}{2} \]
Step 4: Use coefficient of \(\vec{b}\).
\[ \vec{a}\cdot\vec{c}=\frac{1}{2} \Rightarrow \cos\theta_2=\frac{1}{2} \Rightarrow \theta_2=\frac{\pi}{3} \]
Final Answer:
\[ \boxed{\theta_1=\frac{\pi}{2},\theta_2=\frac{\pi}{3}} \]