Question

A unit vector perpendicular to both the vectors $\hat{i}+\hat{j}$ and $\hat{j}+\hat{k}$ is

• A. $\frac{-\hat{i}-\hat{j}+\hat{k}}{\sqrt{3}}$
• B. $\frac{\hat{i}+\hat{j}-\hat{k}}{3}$
• C. $\frac{\hat{i}+\hat{j}+\hat{k}}{\sqrt{3}}$
• D. $\frac{\hat{i}-\hat{j}+\hat{k}}{\sqrt{3}}$

Answer 1

Answer: (D)

$\frac{\hat{i}-\hat{j}+\hat{k}}{\sqrt{3}}$

let vector perpendicular to both is $ai+bj+ck$

Dot product should be zero if 2 vectors are perpendicular.

$(i+j).(ai+bj+ck)=0$

$a+b=0$

$(j+k).(ai+bj+ck)=0$

$b+c=0$

So $a=-b$

$b=b$

$c=-b$

Ratio $a:b:c$ is $-1:1:-1$ or $1:-1:1$

Option D satisfies the above condition. so correct answer is option D.