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(A) Let the vertices of the right-angled triangle be $A$,$B$,and $C$,where $A$ is the vertex with the right angle. Given $\vec{A} = -3\hat{i} + 5\hat{

Vedclass · Accepted Answer

$3\hat{i} + 3\hat{j} + 4\hat{k}$

Vedclass · Answer

(A) Let the vertices of the right-angled triangle be $A$,$B$,and $C$,where $A$ is the vertex with the right angle. Given $\vec{A} = -3\hat{i} + 5\hat{j} + 2\hat{k}$. The midpoint of the hypotenuse $BC$ is $\vec{M} = \frac{\vec{B} + \vec{C}}{2} = 6\hat{i} + 2\hat{j} + 5\hat{k}$. The centroid $\vec{G}$ of a triangle is given by $\vec{G} = \frac{\vec{A} + \vec{B} + \vec{C}}{3}$. We can rewrite this as $\vec{G} = \frac{\vec{A} + 2(\frac{\vec{B} + \vec{C}}{2})}{3} = \frac{\vec{A} + 2\vec{M}}{3}$. Substituting the given vectors: $\vec{G} = \frac{(-3\hat{i} + 5\hat{j} + 2\hat{k}) + 2(6\hat{i} + 2\hat{j} + 5\hat{k})}{3}$ $\vec{G} = \frac{-3\hat{i} + 5\hat{j} + 2\hat{k} + 12\hat{i} + 4\hat{j} + 10\hat{k}}{3}$ $\vec{G} = \frac{9\hat{i} + 9\hat{j} + 12\hat{k}}{3}$ $\vec{G} = 3\hat{i} + 3\hat{j} + 4\hat{k}$. Thus,the correct option is $A$.

In a right-angled triangle,if the position vector of the vertex having the right angle is $\vec{A} = -3\hat{i} + 5\hat{j} + 2\hat{k}$ and the position vector of the midpoint of its hypotenuse is $\vec{M} = 6\hat{i} + 2\hat{j} + 5\hat{k}$,then the position vector of its centroid is

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