Find the coordinates of a point which divides internally the points `(1,3,7),(6,3,2)` in the ratio `2:3.`

Answer

To find the coordinates of the point that divides the line segment joining the points \( A(1, 3, 7) \) and \( B(6, 3, 2) \) in the ratio \( 2:3 \), we can use the section formula.

### Step-by-Step Solution:

1. **Identify the Points and the Ratio**:
- Let \( A(x_1, y_1, z_1) = (1, 3, 7) \)
- Let \( B(x_2, y_2, z_2) = (6, 3, 2) \)
- The ratio \( m:n = 2:3 \)

2. **Use the Section Formula**:
The coordinates of the point \( P(x, y, z) \) that divides the line segment \( AB \) in the ratio \( m:n \) are given by:
\[ x = \frac{mx_2 + nx_1}{m+n}, \quad y = \frac{my_2 + ny_1}{m+n}, \quad z = \frac{mz_2 + nz_1}{m+n} \]

3. **Calculate the x-coordinate**:
\[ x = \frac{2 \cdot 6 + 3 \cdot 1}{2 + 3} = \frac{12 + 3}{5} = \frac{15}{5} = 3 \]

4. **Calculate the y-coordinate**:
\[ y = \frac{2 \cdot 3 + 3 \cdot 3}{2 + 3} = \frac{6 + 9}{5} = \frac{15}{5} = 3 \]

5. **Calculate the z-coordinate**:
\[ z = \frac{2 \cdot 2 + 3 \cdot 7}{2 + 3} = \frac{4 + 21}{5} = \frac{25}{5} = 5 \]

6. **Combine the Coordinates**:
Therefore, the coordinates of the point \( P \) that divides the segment \( AB \) in the ratio \( 2:3 \) are:
\[ P(3, 3, 5) \]

### Final Answer:
The coordinates of the point are \( (3, 3, 5) \).