Question

If five consecutive sides of a regular hexagon represent five unit vectors acting in the same sense, find their resultant vector. (taking first side on x-axis)

Answer 1

$\because$ Regular hexagon is a closed figure.

$\Rightarrow \overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{CD}+\overrightarrow{DE}+\overrightarrow{EF}+\overrightarrow{FA}=0$

$\therefore \overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{CD}+\overrightarrow{DE}+\overrightarrow{EF}=-\overrightarrow{FA}$

$\therefore$ The resultant vector of five consecutive sides=$-\overrightarrow{FA}$

$\therefore$ Magnitude=$1$

Angle between any two consecutive sides=$\frac{(2n-4)\times 90}{n}=\frac{8\times 90}{6}=120$

Unit vector along $\overrightarrow{FA}$ is

$=\cos 30\hat{i}-\sin 60\hat{j}=\frac{1}{2}\hat{i}-\frac{\sqrt{3}}{2}\hat{j}=\frac{1}{2}(\hat{i}-\sqrt{3}\hat{j})$

which is resultant vector