Question

Find the volume of the parallelepiped whose co terminal edges are $4\hat{i}+3\hat{j}+\hat{k},5\hat{i}+9\hat{j}+19\hat{k}$ and $8i+6j+5k$.

Answer 1

$a=4\hat{i}+3\hat{j}+\hat{k}$

$a=5\hat{i}+9\hat{j}+19\hat{k}$

$a=8\hat{i}+6\hat{j}+5\hat{k}$

Volume $=\overrightarrow{a}.(\overrightarrow{b}\times \overrightarrow{c})=\overrightarrow{b}\times \overrightarrow{c}=\left|\begin{array}{ccc}i& j& k\\ 5& 9& 19\\ 8& 6& 6\end{array}\right|$

$i(9\times 5-19\times 6)-j(5\times 5-19\times 8)$

$+k(5\times 6-9\times 8)$

$-69\hat{i}+127\hat{j}-42\hat{k}$

$\overrightarrow{a}(\overrightarrow{b}\times \overrightarrow{c})=\left(4\right)(-69)+3\left(127\right)+\left(1\right)(-42)$

Volume $=63units$