Question

Assertion :If $\overrightarrow{a}=5\hat{i}-\hat{j}+2\hat{k}$, then a vector having magnitude of $8$ units along $\overrightarrow{a}$ is $\frac{8}{\sqrt{30}}(5\hat{i}-\hat{j}+2\hat{k})$ Reason: Vector having modulus m along a given vector ${}^{\prime }{a}^{\prime }$ is given by, $m$ $\times \hat{a}$

• 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

The given vector is $\overrightarrow{a}=5\hat{i}-\hat{j}+2\hat{k}$
$|\overrightarrow{a}|=\sqrt{5^2+(-1{)}^2+2^2}=\sqrt{25+1+4}=\sqrt{30}$
$\therefore$ Unit vector in the direction of vector
$\overrightarrow{a}=\frac{1}{\sqrt{30}}(5\hat{i}-\hat{j}+2\hat{k})$
$\therefore$ Vector of magnitude 8 in the direction of vector $\overrightarrow{a}$
$=8\times \overrightarrow{a}=8\frac{1}{\sqrt{30}}(5\hat{i}-\hat{j}+2\hat{k})$
$=\frac{8}{\sqrt{30}}(5\hat{i}-\hat{j}+2\hat{k})=\frac{40}{\sqrt{30}}\hat{i}-\frac{8}{\sqrt{30}}\hat{j}+\frac{16}{\sqrt{30}}\hat{k}$