Find a unit vector perpendicular to each of the vector $\overrightarrow{\mathrm{a}}\text{ and }\overrightarrow{\mathrm{b}}\text{ where }\overrightarrow{\mathrm{a}}=5\widehat{\mathrm{ı}}+6\widehat{\mathrm{ȷ}}-$$2\widehat{\mathrm{k}}\text{ and }\overrightarrow{\mathrm{b}}=7\widehat{\mathrm{ı}}+6\widehat{\mathrm{ȷ}}+2\widehat{\mathrm{k}}$

OR

Find the volume of the parallelopiped whose adjacent edges are represented by $2\overrightarrow{\mathrm{a}},-\overrightarrow{\mathrm{b}}\text{ and }3\overrightarrow{\mathrm{c}}$ where 
$\overrightarrow{\mathrm{a}}=\widehat{\mathrm{ı}}-\widehat{\mathrm{ȷ}}+2\widehat{\mathrm{k}}$
$\begin{array}{l}\overrightarrow{\mathrm{b}}=3\widehat{\mathrm{ı}}+4\widehat{\mathrm{ȷ}}-5\widehat{\mathrm{k}},\text{ and }\\ \overrightarrow{\mathrm{c}}=2\widehat{\mathrm{ı}}-\widehat{\mathrm{ȷ}}+3\widehat{\mathrm{k}}\end{array}$

First, we will find $\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}$
$\therefore \overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}=\left|\begin{array}{ccc}\widehat{\mathrm{ı}}& \widehat{\mathrm{ȷ}}& \widehat{\mathrm{k}}\\ 5& 6& -2\\ 7& 6& 2\end{array}\right|=(12+12)\widehat{\mathrm{ı}}-(10+14)\mathrm{j}+(30-42)\widehat{\mathrm{k}}=24\widehat{\mathrm{ı}}-24\mathrm{j}-12\widehat{\mathrm{k}}$
Now for unit vector, $|\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}|=\sqrt{{24}^2+(-24{)}^2+(-12{)}^2}=\sqrt{1296}=36$
Therefore, unit vector perpendicular to both  $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}} \mathrm{is},$
$\frac{\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}}{|\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}|}=\frac{24\widehat{i}-24{\mathrm{j}}^{^}-12\widehat{\mathrm{k}}}{36}=\frac{24}{36}\widehat{\mathrm{ı}}-\frac{24}{36}\widehat{\mathrm{ȷ}}-\frac{12}{36}\widehat{\mathrm{k}}=\frac{2}{3}\widehat{\mathrm{ı}}-\frac{2}{3}\widehat{\mathrm{ȷ}}-\frac{1}{3}\widehat{\mathrm{k}}$

OR

Given, 
$\overrightarrow{\mathrm{a}}=\widehat{\mathrm{ı}}-\widehat{\mathrm{ȷ}}+2\widehat{\mathrm{k}}\Rightarrow 2\overrightarrow{\mathrm{a}}=2(\widehat{\mathrm{ı}}-\widehat{\mathrm{ȷ}}+2\widehat{\mathrm{k}})=2\widehat{\mathrm{ı}}-2\widehat{\mathrm{ȷ}}+4\widehat{\mathrm{k}}$
Similarly, $-\overrightarrow{\mathrm{b}}=-3\widehat{\mathrm{ı}}-4\widehat{\mathrm{ȷ}}+5\widehat{\mathrm{k}}\text{ and 3}\overrightarrow{c}=6\widehat{\mathrm{ı}}-3\widehat{\mathrm{ȷ}}+9\widehat{\mathrm{k}}$
Now, Volume of the parallelopiped $=\left|\begin{array}{ccc}2& -2& 4\\ -3& -4& 5\\ 6& -3& 9\end{array}\right|=2(-36-(-15\left)\right)+2(-27-30)+$
$4(9+24)=2(-21)+2(-57)+4\left(33\right)=-42-114+132=|-24|=24$
Therefore, the volume of the parallelopiped is 24.