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

Explanation

To find the coordinates of the point that divides the line segment joining the points \( A(2, 3, -4) \) and \( B(4, -1, 2) \) in the given ratios, we will use the section formula.

### Part (i): Internal Division in the Ratio \( 2:3 \)

1. **Identify the Points and the Ratio**:
- Let \( A(2, 3, -4) \) be \( (x_1, y_1, z_1) \)
- Let \( B(4, -1, 2) \) be \( (x_2, y_2, z_2) \)
- The ratio \( m:n = 2:3 \) (where \( m = 2 \) and \( n = 3 \))

2. **Apply the Section Formula**:
The coordinates \( (x, y, z) \) of the point dividing the line segment in the ratio \( m:n \) internally are given by:
\[ x = \frac{m x_2 + n x_1}{m + n}, \quad y = \frac{m y_2 + n y_1}{m + n}, \quad z = \frac{m z_2 + n z_1}{m + n} \]

3. **Substitute the Values**:
\[ x = \frac{2 \cdot 4 + 3 \cdot 2}{2 + 3} = \frac{8 + 6}{5} = \frac{14}{5} \]
\[ y = \frac{2 \cdot (-1) + 3 \cdot 3}{2 + 3} = \frac{-2 + 9}{5} = \frac{7}{5} \]
\[ z = \frac{2 \cdot 2 + 3 \cdot (-4)}{2 + 3} = \frac{4 - 12}{5} = \frac{-8}{5} \]

4. **Final Coordinates**:
Therefore, the coordinates of the point that divides the line segment internally in the ratio \( 2:3 \) are:
\[ \left( \frac{14}{5}, \frac{7}{5}, \frac{-8}{5} \right) \]

### Part (ii): External Division in the Ratio \( 4:3 \)

1. **Identify the Points and the Ratio**:
- The same points \( A(2, 3, -4) \) and \( B(4, -1, 2) \)
- The ratio \( m:n = 4:3 \) (where \( m = 4 \) and \( n = 3 \))

2. **Apply the Section Formula for External Division**:
The coordinates \( (x, y, z) \) of the point dividing the line segment in the ratio \( m:n \) externally are given by:
\[ x = \frac{m x_2 - n x_1}{m - n}, \quad y = \frac{m y_2 - n y_1}{m - n}, \quad z = \frac{m z_2 - n z_1}{m - n} \]

3. **Substitute the Values**:
\[ x = \frac{4 \cdot 4 - 3 \cdot 2}{4 - 3} = \frac{16 - 6}{1} = 10 \]
\[ y = \frac{4 \cdot (-1) - 3 \cdot 3}{4 - 3} = \frac{-4 - 9}{1} = -13 \]
\[ z = \frac{4 \cdot 2 - 3 \cdot (-4)}{4 - 3} = \frac{8 + 12}{1} = 20 \]

4. **Final Coordinates**:
Therefore, the coordinates of the point that divides the line segment externally in the ratio \( 4:3 \) are:
\[ (10, -13, 20) \]

### Summary of Results:
- Internal Division \( 2:3 \): \( \left( \frac{14}{5}, \frac{7}{5}, \frac{-8}{5} \right) \)
- External Division \( 4:3 \): \( (10, -13, 20) \)