Question

Assertion :Let $\overrightarrow{a}=x\hat{i}-3\hat{j}+7\hat{k}$ and $\overrightarrow{b}=\hat{i}+y\hat{j}-z\hat{k}$ are collinear, then value of $\frac{x{y}^2}{z}=\frac{-9}{7}$ Reason: For linearly dependent vectors coefficients of $\hat{i},\hat{j},\hat{k}$ are in proportion.

• A. Both Assertion & Reason are individually true & Reason is correct explanation of Assertion
• B. Both Assertion & Reason are individually true but Reason is not the correct explanation of Assertion
• C. Assertion is true but Reason is false
• D. Assertion is false but Reason is true

Answer 1

The correct option is A Both Assertion & Reason are individually true & Reason is correct explanation of Assertion

Vectors are collinear

$\therefore \overline{a}=m\overline{b}(m\ne 0)$
$\Rightarrow x\hat{i}+(-3)\hat{j}+7\hat{k}=m\hat{i}+my\hat{j}-mz\hat{k}$

$\Rightarrow \frac{x}{m}=\frac{-3}{my}=\frac{7}{-mz}$

$\Rightarrow xy=-3$ and $xz=-7$ and $3z=7y$

$\Rightarrow x^2{y}^2=9$

$\Rightarrow \frac{x^2{y}^2}{xz}=\frac{-9}{7}$

$\Rightarrow \frac{x{y}^2}{z}=\frac{-9}{7}$
Hence, both assertion and reason are individually true and reason is correct explanation of assertion.

Hence option $A$ is correct.