Join the Most Relevant JEE Main 2025 Test Series & get 99+ percentile! Join Now
Search any question & find its solution
Question: Answered & Verified by Expert
If $\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}$ are three vectors such that $|\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}|=1$, $\overline{\mathrm{c}}=\lambda(\overline{\mathrm{a}} \times \overline{\mathrm{b}})$ and $|\overline{\mathrm{a}}|=\frac{1}{\sqrt{3}},|\overline{\mathrm{b}}|=\frac{1}{\sqrt{2}},|\overline{\mathrm{c}}|=\frac{1}{\sqrt{6}}$, then the angle between $\bar{a}$ and $\bar{b}$ is
MathematicsVector AlgebraMHT CETMHT CET 2023 (14 May Shift 1)
Options:
  • A $\frac{\pi}{6}$
  • B $\frac{\pi}{4}$
  • C $\frac{\pi}{3}$
  • D $\frac{\pi}{2}$
Solution:
2095 Upvotes Verified Answer
The correct answer is: $\frac{\pi}{2}$
Let $\theta$ be the angle between $\bar{a}$ and $\bar{b}$.
$$
\begin{aligned}
& \text { Since } \overline{\mathrm{c}}=\lambda(\overline{\mathrm{a}} \times \overline{\mathrm{b}}) \\
& \Rightarrow \overline{\mathrm{c}} \perp \overline{\mathrm{a}}, \overline{\mathrm{c}} \perp \overline{\mathrm{b}} \\
& \Rightarrow \overline{\mathrm{c}} \cdot \overline{\mathrm{a}}=\overline{\mathrm{c}} \cdot \overline{\mathrm{b}}=0
\end{aligned}
$$

Now,
$$
\begin{aligned}
& |\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}|=1 \\
& \Rightarrow|\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}|^2=1 \\
& \Rightarrow|\overline{\mathrm{a}}|^2+|\overline{\mathrm{b}}|^2+|\overline{\mathrm{c}}|^2+2(\overline{\mathrm{a}} \cdot \overline{\mathrm{b}}+\overline{\mathrm{b}} \cdot \overline{\mathrm{c}}+\overline{\mathrm{c}} \cdot \overline{\mathrm{a}})=1 \\
& \Rightarrow \frac{1}{3}+\frac{1}{2}+\frac{1}{6}+2\{|\overline{\mathrm{a}}||\overline{\mathrm{b}}| \cos \theta\}=1 \\
& \Rightarrow \cos \theta=0 \\
& \Rightarrow \theta=\frac{\pi}{2}
\end{aligned}
$$

Looking for more such questions to practice?

Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.