(i) The coordinates of point R that divides the line segment joining
points $$P(x_1, y_1, z_1)$$ and $$Q(x_2, y_2, z_2)$$ internally in the
ratio $$m:n$$ are
$$\left ( \dfrac{mx_2 + nx_1}{m + n}, \dfrac{my_2 + ny_1}{m + n}, \dfrac{mz_2 + nz_1}{m + n} \right )$$
Let
$$R(x, y, z)$$ be the points that divides the line segment joining
points $$(-2, 3, 5)$$ and $$(1, -4, 6)$$ internally in the ratio $$2 :
3$$
$$x = \dfrac{2(1) + 3(-2)}{2 + 3}, y = \dfrac{2(-4) + 3(3)}{2 + 3}$$ and $$z = \dfrac{2(6) + 3(5)}{2 + 3}$$
i.e., $$x = \dfrac{-4}{5}, y = \dfrac{1}{5},$$ and $$z = \dfrac{27}{5}$$
Thus, the coordinates of the required point are $$\left ( \dfrac{-4}{5}, \dfrac{1}{5}, \dfrac{27}{5} \right )$$
(ii)
The coordinates of point R that divides the line segment joining points
$$P(x_1, y_1, z_1)$$ and $$Q(x_2, y_2, z_2)$$ externally in the ratio
$$m:n$$ are
$$\left ( \dfrac{mx_2 + nx_1}{m - n}, \dfrac{my_2 + ny_1}{m - n}, \dfrac{mz_2 + nz_1}{m - n} \right )$$
Let
$$R(x, y, z)$$ be the point that divides the line segment joining
points $$(-2, 3, 5)$$ and $$(1, -4, 6)$$ externally in the ratio $$2 :
3$$
$$x = \dfrac{2(1) - 3(2)}{2 - 3}, y = \dfrac{2(-4) - 3(3)}{2 - 3},$$ and $$z = \dfrac{2(6) - 3(5)}{2 - 3}$$
i.e., $$x = -8, y = 17,$$ and $$z = 3$$
Thus, the coordinates of the required point are $$(-8, 17, 3).$$