If the vectors vec{AB} = hat{i} + 3hat{j} + 4 | vedclass

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(A) In a triangle $ABC$,if $A$ is the origin,then the position vectors of $B$ and $C$ are $\vec{AB}$ and $\vec{AC}$ respectively. The position vector

Vedclass · Accepted Answer

$\frac{2}{3} \sqrt{22}$

Vedclass · Answer

(A) In a triangle $ABC$,if $A$ is the origin,then the position vectors of $B$ and $C$ are $\vec{AB}$ and $\vec{AC}$ respectively. The position vector of the centroid $G$ is given by $\vec{AG} = \frac{\vec{AB} + \vec{AC}}{3}$.Substituting the given vectors:$\vec{AG} = \frac{(\hat{i} + 3\hat{j} + 4\hat{k}) + (5\hat{i} + \hat{j} + 2\hat{k})}{3}$$\vec{AG} = \frac{6\hat{i} + 4\hat{j} + 6\hat{k}}{3} = 2\hat{i} + \frac{4}{3}\hat{j} + 2\hat{k}$.Now,calculating the magnitude $|\vec{AG}| = \sqrt{2^2 + (\frac{4}{3})^2 + 2^2} = \sqrt{4 + \frac{16}{9} + 4} = \sqrt{8 + \frac{16}{9}} = \sqrt{\frac{72 + 16}{9}} = \sqrt{\frac{88}{9}} = \frac{\sqrt{4 	imes 22}}{3} = \frac{2}{3}\sqrt{22}$.Thus,the correct option is $(a)$.

If the vectors $\vec{AB} = \hat{i} + 3\hat{j} + 4\hat{k}$ and $\vec{AC} = 5\hat{i} + \hat{j} + 2\hat{k}$ are two sides of a triangle $ABC$,whose centroid is $G$,then $|\vec{AG}| = $

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