Question

Find the unit vectors perpendicular to given pair of vectors:$i+j,i-j+k$

• A. $\frac{1}{\sqrt{6}}(i-j-2k),$
• B. $\frac{1}{3}(2i-j-2k),$
• C. $\frac{1}{\sqrt{6}}(i+j+2k),$
• D. $\frac{1}{3}(2i+j-2k),$

Answer 1

Answer: (A)

$\frac{1}{\sqrt{6}}(i-j-2k),$

Let $\overrightarrow{a},\overrightarrow{b}$ be two non-zero vectors. Then $\frac{\overrightarrow{a}\times \overrightarrow{b}}{|\overrightarrow{a}\times \overrightarrow{b}|}$ is a unit vector perpendicular to both $\overrightarrow{a}$ and $\overrightarrow{b}$
Given $i+j,i-j+k$
$(i+j)\times (i-j+k)=-k-j-k+i=i-j-2k$
$\left|\right(i+j)\times (i-j+k\left)\right|=|i-j-2k|=\sqrt{6}$
$\frac{(i+j)\times (i-j+k)}{\left|\right(i+j)\times (i-j+k\left)\right|}=\frac{i-j-2k}{\sqrt{6}}$