Area of a parallelogram with adjacent sides $\overrightarrow a=\overrightarrow p+2\overrightarrow q,$ and $\overrightarrow b=2\overrightarrow p+\overrightarrow q$, where $\overrightarrow p,\overrightarrow q$ are unit vectors with angle $\large\frac{\pi}{6}$ between them is ?

Toolbox:
• Area of parallelogram with adjacent sides $\overrightarrow a$ and $\overrightarrow b$ = $|\overrightarrow a\times\overrightarrow b|$
• $\overrightarrow q\times\overrightarrow p=-\overrightarrow p\times\overrightarrow q$
• $\overrightarrow p\times\overrightarrow q=|\overrightarrow p||\overrightarrow q|sin\theta$
Area of parallelogram = $|\overrightarrow a\times\overrightarrow b|$
$=|(\overrightarrow p+2\overrightarrow q)\times (2\overrightarrow p+\overrightarrow q)|$
$=|2|\overrightarrow p|^2+\overrightarrow p\times\overrightarrow q+4(\overrightarrow q\times \overrightarrow p)+2|\overrightarrow q|^2|$
Given:  $|\overrightarrow p|=|\overrightarrow q|=1$
$\Rightarrow\:$  Area $=|2-3(\overrightarrow p\times\overrightarrow q)+2|$
$=|4-3sin\large\frac{\pi}{6}|$
$=\large\frac{5}{2}$

edited Feb 24, 2014