Question

The angle $\theta$ between the vector $\overline{p}=\hat{i}+\hat{j}+\hat{k}$ 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

The correct option is A ${\cos }^{-1}\left(\frac{1}{\sqrt{3}}\right)$

Vector is given as $\overrightarrow{p}=\hat{i}+\hat{j}+\hat{k}$

Magnitude of the vector $|\overrightarrow{p}|=\sqrt{1^2+1^2+1^2}=\sqrt{3}$

Unit vector in x direction $\overrightarrow{x}=\hat{i}$

Using $\cos \theta =\frac{\overrightarrow{p}.\overrightarrow{x}}{|\overrightarrow{p}||\overrightarrow{x}|}$

$\cos \theta =\frac{(\hat{i}+\hat{j}+\hat{j}).\hat{i}}{\sqrt{3}\times 1}=\frac{1}{\sqrt{3}}$

Angle $\theta =co{s}^{-1}(\frac{1}{\sqrt{3}})$