Question

Two adjacent sides of a parallelogram are $\overrightarrow{a}=$ $\hat{i}-2\hat{j}+3\hat{k}$ and $\overrightarrow{b}=2\hat{i}-4\hat{j}+5\hat{k}$.Find its area.

Answer 1

$\overline{a}+\overline{b}=3\hat{i}-6\hat{j}+8\hat{k}$

$\overline{a}+\overline{b}=\overline{c}\hat{c}=\frac{\overline{c}}{|\overline{c}|}=\frac{3\hat{i}-6\hat{j}+8\hat{k}}{\sqrt{109}}$

$\overline{a}-\overline{b}=\overline{d}=\hat{i}+2\hat{j}-2\hat{k}\hat{d}=\frac{-\hat{i}+2\hat{j}-2\hat{k}}{3}$

Area $=|\overline{a}\times \overline{b}|$

$\overline{a}\times \overline{b}\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& -2& 3\\ 2& -4& 5\end{array}\right|=\hat{i}(-10+12)-\hat{j}(5-6)+\hat{k}(-4+6)$

$=-2\hat{i}+\hat{j}$

Area $=\sqrt{5}$