1. **Stating the problem:** We want to find the coordinates of point P that divides the line segment AB in the ratio $m:n$. 2. **Formula used:** If $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$, and $P$ divides $AB$ such that $AP:PB = m:n$, then the coordinates of $P$ are given by the section formula: $$P = \left( \frac{m x_2 + n x_1}{m + n}, \frac{m y_2 + n y_1}{m + n}, \frac{m z_2 + n z_1}{m + n} \right)$$ 3. **Explanation:** - The ratio $m:n$ means that $P$ divides the segment so that the length from $A$ to $P$ is $m$ parts and from $P$ to $B$ is $n$ parts. - The coordinates of $P$ are weighted averages of the coordinates of $A$ and $B$. 4. **Derivation:** Using the formula for each coordinate: $$x_p = \frac{m x_2 + n x_1}{m + n}$$ $$y_p = \frac{m y_2 + n y_1}{m + n}$$ $$z_p = \frac{m z_2 + n z_1}{m + n}$$ 5. **Intermediate step example:** If we write the $x$ coordinate step by step: $$x_p = \frac{m x_2 + n x_1}{m + n} = \frac{\cancel{m} x_2 + \cancel{n} x_1}{\cancel{m} + \cancel{n}}$$ This shows the weighted sum divided by the total parts. 6. **Summary:** The point $P$ dividing the segment $AB$ in ratio $m:n$ has coordinates: $$\boxed{\left( \frac{m x_2 + n x_1}{m + n}, \frac{m y_2 + n y_1}{m + n}, \frac{m z_2 + n z_1}{m + n} \right)}$$ This formula works for any ratio and any dimension (2D or 3D).