Find the co-ordinates of a point which divides the line segment joining the points `A(2,-1,3)` and `B(4,3,1)` in the ratio `3 : 4` externally.

Explanation

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

### Step-by-Step Solution:

1. **Identify the Points and Ratio:**
- Let \( A(2, -1, 3) \) and \( B(4, 3, 1) \).
- The ratio \( M:N = 3:4 \).

2. **Use the Section Formula for External Division:**
The coordinates \( (x, y, z) \) of the point dividing the line segment externally in the ratio \( M:N \) is given by:
\[ x = \frac{M \cdot x_2 - N \cdot x_1}{M - N} \]
\[ y = \frac{M \cdot y_2 - N \cdot y_1}{M - N} \]
\[ z = \frac{M \cdot z_2 - N \cdot z_1}{M - N} \]
where \( (x_1, y_1, z_1) \) are the coordinates of point A and \( (x_2, y_2, z_2) \) are the coordinates of point B.

3. **Substituting the Values:**
- For \( x \):
\[ x = \frac{3 \cdot 4 - 4 \cdot 2}{3 - 4} = \frac{12 - 8}{-1} = \frac{4}{-1} = -4 \]
- For \( y \):
\[ y = \frac{3 \cdot 3 - 4 \cdot (-1)}{3 - 4} = \frac{9 + 4}{-1} = \frac{13}{-1} = -13 \]
- For \( z \):
\[ z = \frac{3 \cdot 1 - 4 \cdot 3}{3 - 4} = \frac{3 - 12}{-1} = \frac{-9}{-1} = 9 \]

4. **Final Coordinates:**
Therefore, the coordinates of the point that divides the line segment externally in the ratio \( 3:4 \) are:
\[ (-4, -13, 9) \]

### Summary:
The coordinates of the point that divides the line segment joining \( A(2, -1, 3) \) and \( B(4, 3, 1) \) externally in the ratio \( 3:4 \) are \( (-4, -13, 9) \).