Question

Assertion :If $\overrightarrow{v}=\hat{i}+\hat{j}+\hat{k}$ and $\overrightarrow{b}=2\overrightarrow{i}+2\overrightarrow{j}$, then $\left(Pro{j}_{b}v\right)=\hat{i}+\hat{j}$ and component of $\overrightarrow{v}$ orthogonal to $\overrightarrow{b}=-\hat{j}+\hat{i}.$ Reason: $Pro{j}_{b}v=\frac{\overrightarrow{v}\cdot \overrightarrow{b}}{{\left|\overrightarrow{b}\right|}^2}\overrightarrow{b}$ and component of $\overrightarrow{v}$ orthogonal to $\overrightarrow{b}$ is $\overrightarrow{v}-Pro{j}_{b}v$

• 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. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Given

$\overrightarrow{v}=\hat{i}+\hat{j}+\hat{k}$

$\overrightarrow{b}=2\hat{i}+2\hat{j}$

$\left(Pro{j}_{b}v\right)=\hat{i}+\hat{j}$

$componentof\overrightarrow{v}orthogonalto\overrightarrow{b}=-\hat{j}+\hat{i}$

$(\hat{i}+\hat{j}+\hat{k})\cdot (-\hat{j}+\hat{i})=-1+1=0$

$\left(Pro{j}_{b}v\right)=\frac{\overrightarrow{v}\cdot \overrightarrow{b}}{{\left|\overrightarrow{b}\right|}^2}\overrightarrow{b}$

$\left(Pro{j}_{b}v\right)=\frac{(\hat{i}+\hat{j}+\hat{k})\cdot (2\hat{i}+2\hat{j})}{(\sqrt{2^2+2^2}{)}^2}\overrightarrow{b}$

$\left(Pro{j}_{b}v\right)=\frac{2+2}{8}(2\hat{i}+2\hat{j})$

$\left(Pro{j}_{b}v\right)=\frac{4}{8}(2\hat{i}+2\hat{j})$

$\left(Pro{j}_{b}v\right)=\hat{i}+\hat{j}$