Question

Assertion :$\left(A\right):\overline{GA}+\overline{GB}+\overline{GC}=\overline{0}$ where $G$ is the centroid of triangle ABC. Reason: $\left(R\right):\overline{AB}=$ P.V of $B-$ P.V of $A.$

• A. Both Assertion & Reason are individually true & Reason is correct explanation of Assertion
• B. Both Assertion & Reason are individually true but Reason is not the correct explanation of Assertion
• C. Assertion is true but Reason is false
• D. Assertion is false but Reason is true

Answer 1

The correct option is A Both Assertion & Reason are individually true & Reason is correct explanation of Assertion

Let the position vectors of the vertices of the triangle $ABC$ be $\overline{b}$ respectively so that the position vector of
$G=\frac{\overline{a}+\overline{b}+\overline{c}}{3}$

Now $\overline{GA}=$P.V. of $A-$ P.V. of $G$

$=\overline{a}-\frac{\overline{a}+\overline{b}+\overline{c}}{3}=\frac{2\overline{a}-\overline{b}-\overline{c}}{3}$

Similarly

$\overline{GB}=$ P.V. of $B-$ P.V.of $G=\frac{2\overline{b}-\overline{a}-\overline{c}}{3}$

and $\overline{GC}=$P.V. of $C-$ P.V of $G=\frac{2\overline{c}-\overline{a}-\overline{b}}{3}$

$\therefore \overline{GA}+\overline{GB}+\overline{GC}=\frac{1}{3}[(2\overline{a}+2\overline{b}+2\overline{c})-2(\overline{a}+\overline{b}+\overline{c})]=\overline{0}$

$\Rightarrow$ Assertion $\left(A\right)$ is correct obtained by the

Reason $\left(R\right)$ Hence Assertion $\left(A\right)$ & Reason $\left(R\right)$ both are correct &Reason
$\left(R\right)$ is the correct explanation of the Assertion $\left(A\right).$