Search any question & find its solution
Question:
Answered & Verified by Expert
If $\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}$ are unit vectors and $\theta$ is angle between $\overline{\mathrm{a}}$ and $\overline{\mathrm{c}}$ and $\overline{\mathrm{a}}+2 \overline{\mathrm{b}}+2 \overline{\mathrm{c}}=\overline{0}$, then $|\overline{\mathrm{a}} \times \overline{\mathrm{c}}|=$
Options:
Solution:
2906 Upvotes
Verified Answer
The correct answer is:
$\frac{\sqrt{15}}{4}$
$$
\begin{aligned}
& \overline{\mathrm{a}}+2 \overline{\mathrm{b}}+2 \overline{\mathrm{c}}=\overline{0} \\
& \Rightarrow \mathrm{a}+2 \overline{\mathrm{c}}=-2 \overline{\mathrm{b}}
\end{aligned}
$$
Squaring on both sides, we get
$$
\begin{aligned}
& |\overline{\mathrm{a}}|^2+4 \overline{\mathrm{a}} \cdot \overline{\mathrm{c}}+4|\overline{\mathrm{c}}|^2=4|\overline{\mathrm{b}}|^2 \\
& \Rightarrow 1+4|\overline{\mathrm{a}}||\overrightarrow{\mathrm{c}}| \cos \theta+4=4 \\
& \Rightarrow \cos \theta=-\frac{1}{4} \\
& \Rightarrow \sin \theta=\frac{\sqrt{15}}{4} \\
& |\overline{\mathrm{a}} \times \overline{\mathrm{c}}|=|\overline{\mathrm{a}}||\overline{\mathrm{c}}| \sin \theta \\
& \quad=(1)(1)\left(\frac{\sqrt{15}}{4}\right)=\frac{\sqrt{15}}{4}
\end{aligned}
$$
\begin{aligned}
& \overline{\mathrm{a}}+2 \overline{\mathrm{b}}+2 \overline{\mathrm{c}}=\overline{0} \\
& \Rightarrow \mathrm{a}+2 \overline{\mathrm{c}}=-2 \overline{\mathrm{b}}
\end{aligned}
$$
Squaring on both sides, we get
$$
\begin{aligned}
& |\overline{\mathrm{a}}|^2+4 \overline{\mathrm{a}} \cdot \overline{\mathrm{c}}+4|\overline{\mathrm{c}}|^2=4|\overline{\mathrm{b}}|^2 \\
& \Rightarrow 1+4|\overline{\mathrm{a}}||\overrightarrow{\mathrm{c}}| \cos \theta+4=4 \\
& \Rightarrow \cos \theta=-\frac{1}{4} \\
& \Rightarrow \sin \theta=\frac{\sqrt{15}}{4} \\
& |\overline{\mathrm{a}} \times \overline{\mathrm{c}}|=|\overline{\mathrm{a}}||\overline{\mathrm{c}}| \sin \theta \\
& \quad=(1)(1)\left(\frac{\sqrt{15}}{4}\right)=\frac{\sqrt{15}}{4}
\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.