Question

The angle $\theta$ between the vector $p=\hat{i}+\hat{j}+\hat{k}$ and unit vector along $x$-axis is

• A. ${\cos }^{-1}\left(\frac{1}{\sqrt{3}}\right)$
• B. ${\cos }^{-1}\left(\frac{1}{\sqrt{2}}\right)$
• C. ${\cos }^{-1}\left(\frac{\sqrt{3}}{2}\right)$
• D. ${\cos }^{-1}\left(\frac{1}{2}\right)$

Answer 1

Answer: (A)

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

The angle between $p=\hat{i}+\hat{j}+\hat{k}$, $x$-axis and $x=\hat{i}$ is given by:
$\cos \theta =\frac{p\cdot x}{\left|p\right|\left|x\right|}=\frac{(\hat{i}+\hat{j}+\hat{k})\left(\hat{i}\right)}{\sqrt{1^2+1^2+1^2}\cdot \sqrt{1^2}}=\frac{1}{\sqrt{3}}$
$\Rightarrow \theta ={\cos }^{-1}\left(\frac{1}{\sqrt{3}}\right)$