Question

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

• A. $3\mathbf{i} + 3\mathbf{j} + 4\mathbf{k}$
• B. $3\mathbf{i} + 3\mathbf{j} + 3\mathbf{k}$
• C. $\frac{3\mathbf{i} + 7\mathbf{j} + 7\mathbf{k}}{2}$
• D. $4\mathbf{j} + 3\mathbf{k}$

Hint:

The centroid’s position vector is the average of the vertices’ vectors. If the midpoint of a side is given, use $\mathbf{B} + \mathbf{C} = 2\mathbf{M}$ to simplify calculations.

Answer 1

The Correct Option is A

Let the right-angled vertex be $A(-3, 5, 2)$, and the midpoint of the hypotenuse $BC$ be $M(6, 2, 5)$. The centroid $G$ of triangle $ABC$ is: \[ \mathbf{G} = \frac{\mathbf{A} + \mathbf{B} + \mathbf{C}}{3} \] Since $M$ is the midpoint of $BC$, $\mathbf{B} + \mathbf{C} = 2\mathbf{M} = 2 (6\mathbf{i} + 2\mathbf{j} + 5\mathbf{k}) = 12\mathbf{i} + 4\mathbf{j} + 10\mathbf{k}$. Thus: \[ \mathbf{G} = \frac{(-3\mathbf{i} + 5\mathbf{j} + 2\mathbf{k}) + (12\mathbf{i} + 4\mathbf{j} + 10\mathbf{k})}{3} = \frac{(12 - 3)\mathbf{i} + (5 + 4)\mathbf{j} + (2 + 10)\mathbf{k}}{3} \] \[ = \frac{9\mathbf{i} + 9\mathbf{j} + 12\mathbf{k}}{3} = 3\mathbf{i} + 3\mathbf{j} + 4\mathbf{k} \] Option (1) is correct. Options (2), (3), and (4) do not match the computed vector.