Question

Let $a=2i+j+3k$ and $b=i+3j+2k$. Then the volume of the parallelopiped having conterminous edges as $a,b$ and $c$, where $c$ is the vector perpendicular to the plane of $a,b$ and $\left|c\right|=2$ is

Answer 1

for $\overline{c}\to \overline{c}=\frac{\overline{a}\times \overline{b}}{|\overline{a}\times \overline{b}}\times 2$

$\overline{a}\times \overline{b}\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 2& 1& 3\\ 1& 3& 2\end{array}\right|=\hat{i}(2-9)-\hat{j}(4-3)+\hat{k}(6-1)$

$=7\hat{i}-\hat{j}+5\hat{k}$

$\overline{c}=\frac{-7\hat{i}-\hat{j}+5\hat{k}}{\sqrt{75}}\times 2=\frac{-14\hat{i}-2\hat{j}+10\hat{k}}{\sqrt{75}}$

Volume $=\left|\begin{array}{ccc}2& 1& 3\\ 1& 3& 2\\ -14/\sqrt{75}& -2/\sqrt{75}& 10/\sqrt{75}\end{array}\right|$

$=\frac{1}{\sqrt{75}}\left|\begin{array}{ccc}2& 1& 3\\ 1& 3& 2\\ -14& -2& 10\end{array}\right|$

$=\frac{1}{\sqrt{75}}\left(2\right(30+4)-1(10+28)+3(-2+42\left)\right)$

$\frac{1}{\sqrt{75}}(68-38+120)$