Question

If $\hat{i},\hat{j}$ and $\hat{k}$ are unit vectors along x,y and z axes respectively. the angle $\theta$ between the vector $\hat{i}+\hat{j}+\hat{k}$ and vector $\hat{i}$

• A. $\theta =co{s}^{-1}\left(\frac{1}{\sqrt{3}}\right)$
• B. $\theta =si{n}^{-1}\left(\frac{1}{\sqrt{3}}\right)$
• C. $\theta =co{s}^{-1}\left(\frac{\sqrt{3}}{2}\right)$
• D. $\theta =si{n}^{-1}\left(\frac{\sqrt{3}}{2}\right)$

Answer 1

The correct option is A $\theta =co{s}^{-1}\left(\frac{1}{\sqrt{3}}\right)$

$cos\theta =\frac{(\hat{i}+\hat{j}+\hat{k}).\hat{i}}{({\hat{i}}^2+{\hat{j}}^2+{\hat{k}}^2{)}^{\frac{1}{2}}}=\frac{\hat{i}.\hat{i}+\hat{j}.\hat{i}+\hat{k}.\hat{i}}{(1+1+1{)}^{\frac{1}{2}}}$

$\frac{1+0+0}{\sqrt{3}}=\frac{1}{\sqrt{3}}$

$(\because \hat{j}.\hat{i}=\hat{k}.\hat{i}=0and\hat{i}.\hat{i}=1)$

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