Question

Let $i,j,k$ be unit orthonormal vectors and $a$ be any vector, if $a\times r=j$, then $a.r$ is

• A. 0
• B. 1
• C. -1
• D. arbitrary scalar

Answer 1

Answer: (D)

arbitrary scalar

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

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

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

Hence $a.r$ is an arbitrary constant.