Question

If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are mutually perpendicular vectors of equal magnitude, then the angle between $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}$ and $\overrightarrow{a}$ is

• A. ${\cos }^{-1}\left(\frac{1}{3}\right)$
• B. ${\cos }^{-1}\left(\frac{1}{\sqrt{3}}\right)$
• C. $0$
• D. None of these

Answer 1

Answer: (B)

${\cos }^{-1}\left(\frac{1}{\sqrt{3}}\right)$

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

$\therefore (\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}).\overrightarrow{a}=|\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}|\left|\overrightarrow{a}\right|\cos \theta \Rightarrow (\hat{i}+\hat{j}+\hat{k}).\hat{i}=|\hat{i}+\hat{j}+\hat{k}|\left|\hat{i}\right|\cos \theta \Rightarrow 1=\sqrt{3}\cos \theta$

$\Rightarrow \cos \theta =\frac{1}{\sqrt{3}}\Rightarrow \theta ={\cos }^{-1}\left(\frac{1}{\sqrt{3}}\right)$