Question

Assertion :A component of vector $\overrightarrow{b}=4\hat{i}+2\hat{j}+3\hat{k}$ in the direction perpendicular to the direction of vector $\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k}$ is $\hat{i}-\hat{j}$. Reason: A component of vector in the direction of $\hat{a}=\hat{i}+\hat{k}$ is $2\hat{i}+2\hat{j}+2\hat{k}$.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is C Assertion is correct but Reason is incorrect

Component of $\overrightarrow{b}$ in direction of $\overrightarrow{a}$ is $|\overrightarrow{b}|\cos \theta \hat{a}$ where $\theta$ is the angle between $\overrightarrow{a}$ and $\overrightarrow{b}.$
$\cos \theta =\frac{4+2+3}{\sqrt{72}}$
Thus, component along $\overrightarrow{a}=\left(\sqrt{29}\right)\frac{9}{\sqrt{87}}\frac{\hat{i}+\hat{j}+\hat{k}}{\sqrt{3}}=3(\hat{i}+\hat{j}+\hat{k})$
So, component in perpendicular direction of $\hat{a}$ is $\hat{b}-3(\hat{i}+\hat{j}+\hat{k})=\hat{i}-\hat{j}.$