Question

Find the unit vectors orthogonal to both $(2,3,5)$ and $(2,-1,4)$.

Answer 1

Taking $\overline{x}=(2,3,5)$
and $\overline{y}=(2,-1,4)$
$\therefore \overline{x}\times \overline{y}=\left|\begin{array}{ccc}i& j& k\\ 2& 3& 5\\ 2& -1& 4\end{array}\right|$
$=i(12+5)-j(8-10)+k(-2-6)$
$=17i+2j-8k$
$=(17,2,-8)$
$\therefore |\overline{x}\times \overline{y}|=\sqrt{289+4+64}$
$=\sqrt{357}$
$\therefore$ Unit perpendicular vector to $\overline{x}$ and $\overline{y}$
$=\pm \frac{(\overline{x}\times \overline{y})}{|\overline{x}\times \overline{y}|}$
$=\pm \frac{1}{\sqrt{357}}(17,2,-8)$.