Question

Find the volume of a parallelopiped whose sides are given by $-3\hat{i}+7\hat{j}+5\hat{k},-5\hat{i}+7\hat{j}-3\hat{k},7\hat{i}-5\hat{j}-3\hat{k}$.

Answer 1

Let $\overrightarrow{a}=-3\hat{i}+7\hat{j}+5\hat{k},\overrightarrow{b}=-5\hat{i}+7\hat{j}-3\hat{k},\overrightarrow{c}=7\hat{i}-5\hat{j}-3\hat{k}$

Therefore,

Volume of parallelopiped = $\left|\begin{array}{ccc}-3& 7& 5\\ -5& 7& -3\\ 7& -5& -3\end{array}\right|$

$=|-3(-21-15)-7(15+21)+5(25-49)|=|-264|=264cu.units$