Question

Let $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ be the position vectors of three vertices A, B, C of a,triangle respectively. Then the area of this triangle is given by

• A. $\overrightarrow{a}\times\overrightarrow{b}+\overrightarrow{b}\times\overrightarrow{c}+\overrightarrow{c}\times\overrightarrow{a}$
• B. $\frac{1}{2}(\overrightarrow{a}\times\overrightarrow{b}).\overrightarrow{c}$
• C. $\frac{1}{2}|\overrightarrow{a}\times\overrightarrow{b}+\overrightarrow{b}\times\overrightarrow{c}+\overrightarrow{c}\times\overrightarrow{a}|$
• D. none of these.

Answer 1

Answer: (C)

Area of triangle $=\frac{1}{2}\left|\overrightarrow{AB}\times\overrightarrow{AC}\right|$
$=\frac{1}{2}\left|\left(\vec{b}-\vec{a}\right)\times\left(\vec{c}-\vec{a}\right)\right|$
$=\frac{1}{2}\left(\vec{b}\times\vec{c}+\vec{c}\times\vec{a}+\vec{a}\times\vec{b}\right)$