Here are two parallelograms, defined by the vectors $\mathbf{p}$ and $\mathbf{q}$. Can you find their areas?

$$\mathbf{p}=\left(\begin{array}{c}3\\ 0\end{array}\right), \mathbf{q}=\left(\begin{array}{c}5 \\ 2\end{array}\right)$$

$$\mathbf{p}=\left(\begin{array}{c}3 \\ 2\end{array}\right), \mathbf{q}=\left(\begin{array}{c}0 \\ 4\end{array}\right)$$

Select different vectors $\mathbf{p}$ and $\mathbf{q}$ and find the areas of the corresponding parallelograms. Can you discover a quick way of doing this?

Here are two more parallelograms, again defined by vectors $\mathbf{p}$ and $\mathbf{q}$. This time, neither $\mathbf{p}$ nor $\mathbf{q}$ lies along an axis. Can you find the areas of these parallelograms?

$$\mathbf{p}=\left(\begin{array}{c}4 \\ 1\end{array}\right), \mathbf{q}=\left(\begin{array}{c}3 \\ 3\end{array}\right)$$

$$\mathbf{p}=\left(\begin{array}{c}2 \\ 4\end{array}\right), \mathbf{q}=\left(\begin{array}{c}-1 \\ 3\end{array}\right)$$

Try some others.

Now try to generalise this.

Can you find the area of the parallelogram defined by the vectors $p = (\begin{matrix} a \\ b \end{matrix})$ and $q = (\begin{matrix} c \\ d \end{matrix})$ ?