Q53 / 110 · VITEEE 2022 · Mathematics
If \(\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}\) are three unit vectors such that \(\overrightarrow{\mathrm{b}}\) is not parallel to \(\overrightarrow{\mathrm{c}}\) and \(\overrightarrow{\mathrm{a}} \times(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}})=\frac{1}{2} \overrightarrow{\mathrm{b}}\), then the angle between \(\overrightarrow{\mathrm{a}}\) and \(\overrightarrow{\mathrm{c}}\) is
- Correct answerA\(\frac{\pi}{3}\)
- B\(\frac{\pi}{2}\)
- C\(\frac{\pi}{6}\)
- D\(\frac{\pi}{4}\)
Solution
It is given that \(\overrightarrow{\mathrm{a}} \times(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}})=\frac{1}{2} \overrightarrow{\mathrm{b}}\)
Note: \(\overrightarrow{\mathrm{a}} \times(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}})=(\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{c}}) \overrightarrow{\mathrm{b}}-(\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}) \overrightarrow{\mathrm{c}}\)
\(\begin{aligned}
& \Rightarrow \quad \frac{1}{2} \overrightarrow{\mathrm{b}}=(\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{c}}) \overrightarrow{\mathrm{b}}-(\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}) \overrightarrow{\mathrm{c}} \\
& \Rightarrow \quad\left(\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{c}}-\frac{1}{2}\right) \overrightarrow{\mathrm{b}}-(\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}) \overrightarrow{\mathrm{c}}=0 \quad ...(i)
\end{aligned}\)
Since \(\bar{b}\) and \(\bar{c}\) are non-parallel, therefore for the existence of relation (i), the coeff. of \(\bar{b}\) and \(\overline{\mathrm{c}}\) should vanish separately. Therefore, we get \(\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{c}}-\frac{1}{2}=0\) i.e., \(\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{c}}=\frac{1}{2}\) and \(\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}=0\) Since \(\vec{a}, \vec{b}, \vec{c}\) are the unit vectors therefore
\(\overline{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}=\frac{1}{2} \Rightarrow \theta=\frac{\pi}{3} \cdot\left(\because \cos ^{-1}\left(\frac{1}{2}\right)=\theta\right)\)
Hence, the correct answer is option: 'A'.