Question

If $\overrightarrow{p}\mathrm{and}\overrightarrow{q}$are unit vectors forming an angle of 30°; find the area of the parallelogram having $\overrightarrow{a}=\overrightarrow{p}+2\overrightarrow{q}\mathrm{and}\overrightarrow{b}=2\overrightarrow{p}+\overrightarrow{q}$as its diagonals.

Answer 1

$$\begin{gathered} \mathrm{Given}: \\ \overrightarrow{a}=\hat{p}+2\stackrel{⏜}{q} \\ \overrightarrow{b}=2\stackrel{⏜}{p}+\hat{q} \\ \overrightarrow{a}\times \overrightarrow{b}=\left(\overrightarrow{p}+2\overrightarrow{q}\right)\times \left(2\overrightarrow{p}+\overrightarrow{q}\right) \\ =2\overrightarrow{p}\times \overrightarrow{p}+\overrightarrow{p}\times \overrightarrow{q}+4\overrightarrow{q}\times \overrightarrow{p}+2\overrightarrow{q}\times \overrightarrow{q} \\ =2\left(0\right)+\overrightarrow{p}\times \overrightarrow{q}-4\overrightarrow{p}\times \overrightarrow{q}+2\left(0\right) \\ =-3\overrightarrow{p}\times \overrightarrow{q} \\ \text{Area of the parallelogram}=\frac{1}{2}\left|\overrightarrow{a}\times \overrightarrow{b}\right| \\ =\frac{1}{2}\left|-3\left(\overrightarrow{p}\times \overrightarrow{q}\right)\right| \\ =\frac{3}{2}\left|\overrightarrow{p}\right|\left|\overrightarrow{q}\right|\sin {30}^o \\ =\frac{3}{2}\left(1\right)\left(1\right)\left(\frac{1}{2}\right)\text{(∵}\overrightarrow{p}\text{and}\overrightarrow{q}\text{are unit vectors)} \\ =\frac{3}{4}\text{sq. units} \end{gathered}$$