Question

If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are the vertices of a triangle $ABC$ then $|\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{a}|$ is equal to

• A. Area of the triangle $△ABC$
• B. Two times Area of the triangle $△ABC$
• C. Three times Area of the triangle $△ABC$
• D. Four times Area of the triangle $△ABC$

Answer 1

The correct option is B Two times Area of the triangle $△ABC$

As we know,
Area of triangle having vertices as $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ is $\frac{1}{2}|\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{a}|$
$\Rightarrow |\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{a}|=2\left[\text{Area of}△ABC\right]$