Question

Find the area of the parallelogram formed by the vectors $A=\left[3,2\right]$ and $B=\left[-3,7\right]$

Answer 1

Solve for area of the given parallelogram

Given, $\overrightarrow{A}=3\hat{i}+2\hat{j}$ and $\overrightarrow{B}=-3\hat{i}+7\hat{j}$

We know that area of the parallelogram formed by the vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ $=\left|\overrightarrow{A}\times \overrightarrow{B}\right|$

Now,

$\overrightarrow{A}\times \overrightarrow{B}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 3& 2& 0\\ -3& 7& 0\end{array}\right|$

$$\begin{gathered} =\hat{i}(0-0)-\hat{j}(0-0)+\hat{k}(21+6) \\ =27\hat{k} \end{gathered}$$

$\therefore$Area of the parallelogram$\mathbf{=}\left|\overrightarrow{A}\times \overrightarrow{B}\right|$

$=\left|27\hat{k}\right|$

$=\sqrt{{\left(27\right)}^2}$

$=27$

Hence, the area of the given parallelogram is $27sq.units$