Find the coordinates of the points which divide the join of the points (2-1, 3) and (4,3,1) in the ratio 3:4 internally.

Answer

To find the coordinates of the point that divides the line segment joining the points \( A(2, -1, 3) \) and \( B(4, 3, 1) \) in the ratio \( 3:4 \) internally, we can use the section formula for three-dimensional coordinates.

### Step-by-Step Solution:

1. **Identify the coordinates of the points**:
- Let \( A(x_1, y_1, z_1) = (2, -1, 3) \)
- Let \( B(x_2, y_2, z_2) = (4, 3, 1) \)
- The ratio in which the point divides the line segment is \( m:n = 3:4 \).

2. **Apply the section formula**:
The coordinates \( (x, y, z) \) of the point \( P \) that divides the line segment \( AB \) in the ratio \( m:n \) are given by:
\[ x = \frac{m \cdot x_2 + n \cdot x_1}{m+n} \]
\[ y = \frac{m \cdot y_2 + n \cdot y_1}{m+n} \]
\[ z = \frac{m \cdot z_2 + n \cdot z_1}{m+n} \]

3. **Substitute the values for \( x \)**:
\[ x = \frac{3 \cdot 4 + 4 \cdot 2}{3 + 4} = \frac{12 + 8}{7} = \frac{20}{7} \]

4. **Substitute the values for \( y \)**:
\[ y = \frac{3 \cdot 3 + 4 \cdot (-1)}{3 + 4} = \frac{9 - 4}{7} = \frac{5}{7} \]

5. **Substitute the values for \( z \)**:
\[ z = \frac{3 \cdot 1 + 4 \cdot 3}{3 + 4} = \frac{3 + 12}{7} = \frac{15}{7} \]

6. **Combine the results**:
The coordinates of the point \( P \) that divides the line segment \( AB \) in the ratio \( 3:4 \) are:
\[ P\left(\frac{20}{7}, \frac{5}{7}, \frac{15}{7}\right) \]

### Final Answer:
The required coordinates are \( P\left(\frac{20}{7}, \frac{5}{7}, \frac{15}{7}\right) \).