Question

ABCD is a tetrahedron. \( \vec{i}-2\vec{j}+3\vec{k} \), \( -2\vec{i}+\vec{j}+3\vec{k} \), \( 3\vec{i}+2\vec{j}-\vec{k} \) are the position vectors of the points A, B, C respectively. \( -\vec{i}+2\vec{j}-3\vec{k} \) is the position vector of the centroid of the triangular face BCD. If G is the centroid of the tetrahedron, then GD =

• A. \( \frac{\sqrt{13}}{\sqrt{2}} \)
• B. \( \sqrt{23} \)
• C. \( \frac{\sqrt{213}}{\sqrt{2}} \)
• D. \( \sqrt{46} \)

Hint:

The centroid of a set of points is simply the average of their position vectors. For a triangle with vertices A, B, C, the centroid is \((\vec{a}+\vec{b}+\vec{c})/3\). For a tetrahedron with vertices A, B, C, D, the centroid is \((\vec{a}+\vec{b}+\vec{c}+\vec{d})/4\).

Answer 1

The Correct Option is C

Let the position vectors of the vertices be \( \vec{a}, \vec{b}, \vec{c}, \vec{d} \).
Given: \( \vec{a} = \vec{i}-2\vec{j}+3\vec{k} \), \( \vec{b} = -2\vec{i}+\vec{j}+3\vec{k} \), \( \vec{c} = 3\vec{i}+2\vec{j}-\vec{k} \).
Let \( \vec{p} \) be the position vector of the centroid of face BCD. Given \( \vec{p} = -\vec{i}+2\vec{j}-3\vec{k} \).
The formula for the centroid of triangle BCD is \( \vec{p} = \frac{\vec{b}+\vec{c}+\vec{d}}{3} \).
We can find the position vector of vertex D, \( \vec{d} \), from this relation: \( \vec{d} = 3\vec{p} - \vec{b} - \vec{c} \).
\( \vec{d} = 3(-\vec{i}+2\vec{j}-3\vec{k}) - (-2\vec{i}+\vec{j}+3\vec{k}) - (3\vec{i}+2\vec{j}-\vec{k}) \).
\( \vec{d} = (-3+2-3)\vec{i} + (6-1-2)\vec{j} + (-9-3+1)\vec{k} = -4\vec{i} + 3\vec{j} - 11\vec{k} \).
Now, find the position vector of the centroid of the tetrahedron, \( \vec{g} \).
\( \vec{g} = \frac{\vec{a}+\vec{b}+\vec{c}+\vec{d}}{4} \).
\( \vec{g} = \frac{(\vec{i}-2\vec{j}+3\vec{k}) + (-2\vec{i}+\vec{j}+3\vec{k}) + (3\vec{i}+2\vec{j}-\vec{k}) + (-4\vec{i}+3\vec{j}-11\vec{k})}{4} \).
\( \vec{g} = \frac{(1-2+3-4)\vec{i} + (-2+1+2+3)\vec{j} + (3+3-1-11)\vec{k}}{4} = \frac{-2\vec{i} + 4\vec{j} - 6\vec{k}}{4} = -\frac{1}{2}\vec{i} + \vec{j} - \frac{3}{2}\vec{k} \).
We need to find the distance GD, which is the magnitude of the vector \( \vec{d} - \vec{g} \).
\( \vec{d} - \vec{g} = (-4 - (-\frac{1}{2}))\vec{i} + (3-1)\vec{j} + (-11 - (-\frac{3}{2}))\vec{k} \).
\( \vec{d} - \vec{g} = -\frac{7}{2}\vec{i} + 2\vec{j} - \frac{19}{2}\vec{k} \).
\( GD = |\vec{d} - \vec{g}| = \sqrt{(-\frac{7}{2})^2 + 2^2 + (-\frac{19}{2})^2} = \sqrt{\frac{49}{4} + 4 + \frac{361}{4}} \).
\( GD = \sqrt{\frac{49+16+361}{4}} = \sqrt{\frac{426}{4}} = \sqrt{\frac{213}{2}} = \frac{\sqrt{213}}{\sqrt{2}} \).