Question

$A$: The angle between the vectors $\hat{i}-2\hat{j}+\hat{k},2\hat{i}-\hat{j}-\hat{k}$ is $\frac{\pi }{3}$.
$R$: If $\theta$ is the angle between $\overrightarrow{a},\overrightarrow{b}$ then $\cos \theta =\frac{\overrightarrow{a}.\overrightarrow{b}}{|\overrightarrow{a}||\overrightarrow{b}|}$.

• A. Both $A$and $R$ are true and $R$ is the correct explanation of $A$
• B. Both $A$and $R$ are true and $R$ is the not correct explanation of $A$
• C. $A$ is true but $R$ is false
• D. $A$ is false but $R$ is true

Answer 1

The correct option is A Both $A$and $R$ are true and $R$ is the correct explanation of $A$

Let $\overrightarrow{a}=\hat{i}-2\hat{j}+\hat{k},\overrightarrow{b}=2\hat{i}-\hat{j}-\hat{k}$
$\cos \theta =\frac{\overrightarrow{a}\cdot \overrightarrow{b}}{\left|\overrightarrow{a}\right|\times \left|\overrightarrow{b}\right|}$
$\cos \theta =\frac{2+2-1}{\sqrt{6}\times \sqrt{6}}=\frac{3}{6}=0.5$
$\Rightarrow \theta =\frac{\pi }{3}$