Question

If G is the centroid of a triangle ABC then$\overset{\to }{\mathop{\text{GA}}}\,+\overset{\to }{\mathop{\text{GB}}}\,+\overset{\to }{\mathop{\text{GC}}}\,$ is equal to :
A . $\overset{\to }{\mathop{0}}\,$
B. $3\overset{\to }{\mathop{\text{GA}}}\,$
C. $3\overset{\to }{\mathop{\text{GB}}}\,$
D. $3\overset{\to }{\mathop{\text{GC}}}\,$

Answer 1

Hint: In triangle ABC we can define centroid as $\overrightarrow{G}=\dfrac{\overrightarrow{A}+\overrightarrow{B}+\overrightarrow{C}}{3}$ where $\overrightarrow{A},\overrightarrow{B},\overrightarrow{C}$ are sides vector of triangle ABC. If have length of sides of triangle then we can use it as in same form.

Complete step by step solution:
If sides vectors of triangle ABC are $\overrightarrow{A},\overrightarrow{B},\overrightarrow{C}$. Then we can write centroid of triangle as
$\overrightarrow{G}=\dfrac{\overrightarrow{A}+\overrightarrow{B}+\overrightarrow{C}}{3}$
We can arrange it as
$3\overrightarrow{G}=\overrightarrow{A}+\overrightarrow{B}+\overrightarrow{C}$..................................(i)
We can also write
 $\overrightarrow{GA}=\overrightarrow{A}-\overrightarrow{G}$
$\overrightarrow{GB}=\overrightarrow{B}-\overrightarrow{G}$
$\overrightarrow{GC}=\overrightarrow{C}-\overrightarrow{G}$
Hence given expression can be written as
\[\Rightarrow \overrightarrow{GA}+\overrightarrow{GB}+\overrightarrow{GC}=\overrightarrow{A}-\overrightarrow{G}+\overrightarrow{B}-\overrightarrow{G}+\overrightarrow{C}-\overrightarrow{G}\]
\[\Rightarrow \overrightarrow{GA}+\overrightarrow{GB}+\overrightarrow{GC}=\overrightarrow{A}+\overrightarrow{B}+\overrightarrow{C}-3\overrightarrow{G}\]
From equation (i) we can write value of \[\overrightarrow{A}+\overrightarrow{B}+\overrightarrow{C}\]
\[\Rightarrow \overrightarrow{GA}+\overrightarrow{GB}+\overrightarrow{GC}=3\overrightarrow{G}-3\overrightarrow{G}\]
\[\Rightarrow \overrightarrow{GA}+\overrightarrow{GB}+\overrightarrow{GC}=0\]
Hence option a is correct.

Note: We can use the same formula of centroid in terms of coordinate as well. Also we can write $\overrightarrow{AB}$ as difference of individual vector A and vector B as below:
$\overrightarrow{AB}=\overrightarrow{B}-\overrightarrow{A}$
But we always write vector B first and then vector A. We need to remember this point.