Question

Assertion :If $i,j,k$ are orthonormal unit vectors and $a$ is any vector. If $a\times r=\hat{j},$ then $a.r$ is arbitrary constant. Reason: For any two arbitrary vectors $a$ and $r$, ${|a\times r|}^2+{|a.r|}^2={\left|a\right|}^2{\left|r\right|}^2$

• 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 incorrect but Reason is correct
• 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

we have, ${|a\times r|}^2+{|a.r|}^2={\left|a\right|}^2{\left|r\right|}^2$

$\Rightarrow {\left|j\right|}^2+{(a.r)}^2={\left|a\right|}^2{\left|r\right|}^2$

$\Rightarrow (a.r)=\pm \sqrt{{\left|a\right|}^2{\left|r\right|}^2-1.}$

This shows that $a.r$ depends on $\left|r\right|$ for given $a.$

Hence, $a.r$ is an arbitrary constant.