Question

The angle between two vectors $\overrightarrow{P}=3\hat{i}+2\hat{j}+3\hat{k}$ and $\overrightarrow{Q}=2\hat{i}-\hat{j}+3\hat{k}$ is:

• A. ${\cos }^{-1}[\frac{5}{\sqrt{77}}]$
• B. ${\cos }^{-1}[\frac{\sqrt{77}}{6.5}]$
• C. ${\cos }^{-1}[\frac{6.5}{\sqrt{77}}]$
• D. ${\sin }^{-1}[\frac{\sqrt{77}}{6.5}]$

Answer 1

Answer: (C)

${\cos }^{-1}[\frac{6.5}{\sqrt{77}}]$

$\overrightarrow{P}=3\hat{i}+2\hat{j}+3\hat{k}\overrightarrow{Q}=2\hat{i}-\hat{j}+3\hat{k}\angle \overrightarrow{P}\overrightarrow{Q}=\theta \cos \theta =\frac{\overrightarrow{P}\cdot \overrightarrow{Q}}{\left|\overrightarrow{P}\right|\left|\overrightarrow{Q}\right|}=\frac{3\times 2+2\times (-1)+3\times 3}{\sqrt{3^2+2^2+3^2}\sqrt{2^2+{(-1)}^2+3^2}}=\frac{13}{\sqrt{22}\sqrt{14}}\Rightarrow \theta ={\cos }^{-1}\frac{6.5}{\sqrt{77}}$