Question

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are the two sides of a triangle, then the area of triangle will be given by $|\overrightarrow{a}\times \overrightarrow{b}|$

• A. True
• B. False

Answer 1

The correct option is B False


As shown in the figure if $\overrightarrow{a}$ and $\overrightarrow{b}$ represent the two sides of a triangle then the area of triangle is given by $\frac{1}{2}|\overrightarrow{a}\times \overrightarrow{b}|$ and not $\overrightarrow{a}\times \overrightarrow{b}$
Why is it so?
Look area of a triangle is given by $\frac{1}{2}$(Base) $\times$ ( Perpendicular distance of the base from the opposite vertex)
So $\frac{1}{2}|\overrightarrow{a}\times \overrightarrow{b}|$=$\frac{1}{2}\left(|\overrightarrow{a}||\overrightarrow{b}|sin\varphi \right)$ where$\varphi$ is the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$
Now $|\overrightarrow{b}|$ sin$\varphi$ will be nothing but the perpendicular distance of $\overrightarrow{a}$ from the opposite vertex)
So, $\frac{1}{2}\left(|\overrightarrow{a}||\overrightarrow{b}|sin\varphi \right)$=$\frac{1}{2}\left(|\overrightarrow{a}|\right)$ x perpendicular distance of $\overrightarrow{a}$ from the opposite vertex)
Which is nothing but the area of the triangle.