Question

If $\hat{i},\hat{j}$ and $\hat{k}$ represent unit vectors along the x, y and z-axes respectively, then the angle $\theta$ between the vectors $(\hat{i}+\hat{j}+\hat{k})$ and $(\hat{i}+\hat{j})$ is equal to:

• A. $si{n}^{-1}\left(\frac{1}{\sqrt{3}}\right)$
• B. $si{n}^{-1}\left(\sqrt{\frac{2}{3}}\right)$
• C. $co{s}^{-1}\left(\frac{1}{\sqrt{3}}\right)$
• D. ${90}^o$

Answer 1

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

$(\hat{i}+\hat{j}+\hat{k}).(\hat{i}+\hat{j})=\left(\sqrt{3}\right)\left(\sqrt{2}\right)\cos \theta$

$(1+1)=\left(\sqrt{3}\right)\left(\sqrt{2}\right)\cos \theta$

$\cos \theta =\sqrt{\frac{2}{3}}$

$\theta ={\cos }^{-1}\left(\sqrt{\frac{2}{3}}\right)$

Using formula, ${\cos }^{-1}x={\sin }^{-1}\left(\sqrt{1-x^2}\right)$ we will get,

$\theta ={\sin }^{-1}\left(\sqrt{1-\frac{2}{3}}\right)={\sin }^{-1}\left(\frac{1}{\sqrt{3}}\right)$