Find the position vector of the centroid of the `Delta ABC` when the position vectors of its vertices are `A(1, 3, 0), B(2, 1, 1)` and `C(0,-1, 0)`.

Answer

To find the position vector of the centroid of triangle ABC with vertices A(1, 3, 0), B(2, 1, 1), and C(0, -1, 0), we can follow these steps:

### Step 1: Identify the coordinates of the vertices.
The position vectors of the vertices are given as:
- A = (1, 3, 0)
- B = (2, 1, 1)
- C = (0, -1, 0)

### Step 2: Use the formula for the centroid.
The position vector of the centroid \( G \) of a triangle with vertices \( A(x_1, y_1, z_1) \), \( B(x_2, y_2, z_2) \), and \( C(x_3, y_3, z_3) \) is given by:
\[ G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}, \frac{z_1 + z_2 + z_3}{3} \right) \]

### Step 3: Calculate the x-coordinate of the centroid.
Using the x-coordinates of A, B, and C:
\[ x_G = \frac{1 + 2 + 0}{3} = \frac{3}{3} = 1 \]

### Step 4: Calculate the y-coordinate of the centroid.
Using the y-coordinates of A, B, and C:
\[ y_G = \frac{3 + 1 - 1}{3} = \frac{3}{3} = 1 \]

### Step 5: Calculate the z-coordinate of the centroid.
Using the z-coordinates of A, B, and C:
\[ z_G = \frac{0 + 1 + 0}{3} = \frac{1}{3} \]

### Step 6: Write the position vector of the centroid.
Combining the coordinates calculated:
\[ G = (1, 1, \frac{1}{3}) \]
In vector notation, the position vector of the centroid \( G \) can be expressed as:
\[ \vec{G} = 1 \hat{i} + 1 \hat{j} + \frac{1}{3} \hat{k} \]

### Final Answer:
The position vector of the centroid of triangle ABC is:
\[ \vec{G} = \hat{i} + \hat{j} + \frac{1}{3} \hat{k} \]