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## 'Areas of Parallelograms' printed from http://nrich.maths.org/

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

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

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

Choose different vectors $\mathbf{p}$ and $\mathbf{q}$, where one vector is parallel to an axis,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 $\mathbf{q}$ is parallel to an axis.

Can you find the areas of these parallelograms?

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

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

Choose some other vectors

**p** and

**q**, where neither

**p** nor

**q** is parallel to an axis.

Can you find a quick way of working out the axes of the corresponding parallelograms?

**Can you find the area of the parallelogram defined by the vectors $\mathbf{p}=\left(\begin{array}{c}a \\ b\end{array}\right)$ and $\mathbf{q}=\left(\begin{array}{c}c \\ d\end{array}\right)$?**
If you have found a rule, does it ever give you negative areas?

If so, can you predict which vector pairs have this effect?