Question

The angle between the vectors $(\hat{i}+\hat{j}+\hat{k})$ and $(\hat{i}-\hat{j}-\hat{k})$ is:

• A. $si{n}^{-1}\frac{\sqrt{8}}{3}$
• B. $si{n}^{-1}\frac{1}{3}$
• C. ${\cos }^{-1}\frac{\sqrt{8}}{3}$
• D. ${\cos }^{-1}\sqrt{\frac{8}{3}}$

Answer 1

The correct option is A $si{n}^{-1}\frac{\sqrt{8}}{3}$

$\overrightarrow{A}=\overline{i}+\overline{j}+\overline{k}\overrightarrow{B}=\overline{i}-\overline{j}-\overline{k}\angle \overrightarrow{A}\overrightarrow{B}=\theta \cos \theta =\frac{\overrightarrow{A}\cdot \overrightarrow{B}}{\left|\overrightarrow{A}\right|\left|\overrightarrow{B}\right|}=\frac{-1}{\sqrt{3}\sqrt{3}}=\frac{-1}{3}{\sin }^2\theta +{\cos }^2\theta =1\Rightarrow {\sin }^2\theta =1-\frac{1}{9}=\frac{8}{9}\Rightarrow \sin \theta =\frac{\sqrt{8}}{3}\Rightarrow \theta ={\sin }^{-1}\frac{\sqrt{8}}{3}$