Question

If two vertices of a triangle are $A(3,1,4)$ and $B(-4,5,-3)$ and the centroid of the triangle is $G(-1,2,1),$ then the third vertex C of the triangle is

• A. $(2,0,2)$
• B. $(-2,0,2)$
• C. $(0,-2,2)$
• D. $(2,-2,0)$

Hint:

Logic Tip: The centroid formula works identically for each coordinate component individually. Instead of using full vector notation, you can solve for the $x$, $y$, and $z$ coordinates separately using equations like $x_g = \frac{x_1 + x_2 + x_3}{3}$. For example, solving $-1 = \frac{3 + (-4) + x_3}{3}$ quickly yields $x_3 = -2$.

Answer 1

The Correct Option is B

Concept:
The centroid $G$ of a triangle with vertices $A$, $B$, and $C$ having position vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$ respectively is given by the vector formula: $$\vec{g} = \frac{\vec{a} + \vec{b} + \vec{c{3}$$
Step 1: Identify the given position vectors.
Given the coordinates of vertices $A$ and $B$, and centroid $G$, their corresponding position vectors are: $$\vec{a} = 3\hat{i} + 1\hat{j} + 4\hat{k}$$ $$\vec{b} = -4\hat{i} + 5\hat{j} - 3\hat{k}$$ $$\vec{g} = -\hat{i} + 2\hat{j} + \hat{k}$$
Step 2: Set up the centroid formula to find the third vertex.
Let the position vector of the third vertex $C$ be $\vec{c}$. Using the centroid formula: $$\vec{g} = \frac{\vec{a} + \vec{b} + \vec{c{3}$$ Rearranging the equation to solve for $\vec{c}$: $$3\vec{g} = \vec{a} + \vec{b} + \vec{c}$$ $$\vec{c} = 3\vec{g} - (\vec{a} + \vec{b})$$
Step 3: Substitute the vectors and calculate $\vec{c}$.
First, calculate $3\vec{g}$: $$3\vec{g} = 3(-\hat{i} + 2\hat{j} + \hat{k}) = -3\hat{i} + 6\hat{j} + 3\hat{k}$$ Next, calculate the sum of vectors $\vec{a}$ and $\vec{b}$: $$\vec{a} + \vec{b} = (3 - 4)\hat{i} + (1 + 5)\hat{j} + (4 - 3)\hat{k} = -\hat{i} + 6\hat{j} + \hat{k}$$ Now, substitute these into the rearranged formula to find $\vec{c}$: $$\vec{c} = (-3\hat{i} + 6\hat{j} + 3\hat{k}) - (-\hat{i} + 6\hat{j} + \hat{k})$$ $$\vec{c} = (-3 - (-1))\hat{i} + (6 - 6)\hat{j} + (3 - 1)\hat{k}$$ $$\vec{c} = -2\hat{i} + 0\hat{j} + 2\hat{k}$$
Step 4: Determine the coordinates of vertex C.
From the calculated position vector $\vec{c} = -2\hat{i} + 0\hat{j} + 2\hat{k}$, the coordinates of the third vertex $C$ are: $$C \equiv (-2, 0, 2)$$