Mathematics - Vector Question with Solution | TestHub

We have,

$\left|\overrightarrow{a}\right|=\left|\overrightarrow{b}\right|=\left|\overrightarrow{c}\right|$

$\overrightarrow{a}·\overrightarrow{b}=\overrightarrow{b}·\overrightarrow{c}=\overrightarrow{c}·\overrightarrow{a}=0$

Since, vectors are mutually perpendicular and of equally magnitude, hence let

$\overrightarrow{a}=\hat{i}, \overrightarrow{b}=\hat{j}, \overrightarrow{c}=\hat{k}$

Now,

$\left(\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}\right)·\overrightarrow{a}=\left(\hat{i}+\hat{j}+\hat{k}\right)·\hat{i}$

$\Rightarrow \left(\hat{i}+\hat{j}+\hat{k}\right)·\hat{i}=\sqrt{3}·1·\cos \theta$

$\Rightarrow 1+0+0=\sqrt{3}·1·\cos \theta$

$\Rightarrow \sqrt{3}\cos \theta =1$

$\Rightarrow \cos \theta =\frac{1}{\sqrt{3}}$

$\Rightarrow \cos 2\theta =2{\cos }^2\theta -1=\frac{2}{3}-1$

$\Rightarrow \cos 2\theta =-\frac{1}{3}$

$\Rightarrow {\cos }^{2}2\theta =\frac{1}{9}$

$\Rightarrow 36{\cos }^{2}2\theta =4$