(i) Let P(x, y, z) be any point which divides the line segment joining the points A(–2, 3, 5) and B(1, –4, 6) in the ratio 2 : 3 internally. Then x = $\frac{2 \times 1 + 3 \times (-2)}{2+3}$ = $\frac{2-6}{5}$ = $\frac{-4}{5}$ , y = $\frac{2 \times (-4) + 3 \times 3}{2+3}$ = $\frac{-8+9}{5}$ = $\frac{1}{5}$ z = $\frac{2 \times 6 + 3 \times 5}{2+3}$ = $\frac{12+15}{5}$ = $\frac{27}{5}$ ∴ Coordinates of P are $\left(-\frac{4}{5},\frac{1}{5},\frac{27}{5}\right)$ (ii) Let P(x, y, z) be any point which divides the line segment joining the points A(–2, 3, 5) and B(1, –4, 6) in the ratio 2 : 3 externally. Then x = $\frac{2 \times 1 - 3 \times (-2)}{2-3}$ = $\frac{2+6}{-1}$ = - 8 , y = $\frac{2 \times (-4) - 3 \times 3}{2-3}$ = $\frac{-8-9}{-1}$ = 17 z = $\frac{2 \times 6 - 3 \times 5}{2-3}$ = $\frac{12-15}{-1}$ = 3 ∴ Coordinates of P are (–8, 17, 3)