Question

Find the coordinates of the point which divides the line joining the points \( (2, -5, 1) \) and \( (1, 4, -6) \) internally in the ratio 2 : 3.

Hint:

When dividing a line segment in a given ratio, use the section formula to calculate the coordinates of the dividing point.

Answer 1

Step 1: Formula for the Coordinates of the Dividing Point.
The formula for the coordinates of a point dividing a line segment internally in the ratio \( m : n \) is given by: \[ P = \left( \frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n}, \frac{mz_2 + nz_1}{m + n} \right). \] Let the points be \( A(2, -5, 1) \) and \( B(1, 4, -6) \), and the ratio be \( 2 : 3 \).
Step 2: Apply the Formula.
Using the formula, we find the coordinates of the point \( P \) that divides the line joining \( A \) and \( B \) in the ratio 2:3. \[ P_x = \frac{2 \times 1 + 3 \times 2}{2 + 3} = \frac{2 + 6}{5} = \frac{8}{5}, \] \[ P_y = \frac{2 \times 4 + 3 \times (-5)}{2 + 3} = \frac{8 - 15}{5} = \frac{-7}{5}, \] \[ P_z = \frac{2 \times (-6) + 3 \times 1}{2 + 3} = \frac{-12 + 3}{5} = \frac{-9}{5}. \]
Step 3: Conclusion.
Thus, the coordinates of the point dividing the line internally in the ratio 2:3 are: \[ \left( \frac{8}{5}, \frac{-7}{5}, \frac{-9}{5} \right). \]