Question

If G be the centroid of the triangle ABC, then the value of $\overrightarrow{AG}+\overrightarrow{BG}+\overrightarrow{CG}$ equals ( 0 being origin)

• A. $1$
• B. $3\left(\overrightarrow{OG}\right)$
• C. $0$
• D. $3\left(\overrightarrow{GO}\right)$

Answer 1

The correct option is C $0$

Let us given that,

If G is the centroid of a triangle $ABC$ and let $O$ be the position vector.

Then, let

$OA=\overrightarrow{a}$

$OB=\overrightarrow{b}$

$OC=\overrightarrow{c}$

$OG=\overrightarrow{g}$

We know that, using centroid formula

$\overrightarrow{OG}=\frac{\overrightarrow{OA}\overrightarrow{+OB}\overrightarrow{+OC}}{3}$

$\overrightarrow{g}=\frac{\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}}{3}$

$3\overrightarrow{g}=\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}$

$\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=3\overrightarrow{g}......\left(1\right)$

Now,

According to given question,

$\overrightarrow{GA}+\overrightarrow{GB}+\overrightarrow{GC}$

$=(\overrightarrow{OA}-\overrightarrow{OG})+(\overrightarrow{OB}-\overrightarrow{OG})+(\overrightarrow{OC}-\overrightarrow{OG})$

$=\overrightarrow{OA}+\overrightarrow{OB}+\overrightarrow{OC}-3\overrightarrow{OG}$

$=\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}-3\overrightarrow{g}$ by equation (1)

$=3\overrightarrow{g}-3\overrightarrow{g}=0$

Hence, it is complete solution.

option $C$ is correct.