To solve problems 18, 19, 20, and 21, let's analyze and provide step-by-step solutions. Here's a quick summary of what each problem asks:

Problem 18: Find the maximum and minimum values of $r^2 = a^2 x^2 + b^2 y^2 + c^2 z^2$, subject to the constraints $x^2 + y^2 + z^2 = 1$ and $l x + m y + n z = 0$.

This is a constrained optimization problem where you will need to use the method of Lagrange multipliers to find the extrema. The function to optimize is $r^2 = a^2 x^2 + b^2 y^2 + c^2 z^2$, and the constraints are $g_1(x, y, z) = x^2 + y^2 + z^2 - 1 = 0$ and $g_2(x, y, z) = lx + my + nz = 0$.

Steps:

1. Set up the Lagrange multiplier equations by differentiating the objective function and constraints.
2. Solve the resulting system of equations to find critical points.
3. Analyze the critical points to determine the maximum and minimum values of $r^2$.

Problem 19: Find the maximum and minimum radii vectors of the section of the surface $(x^2 + y^2 + z^2) = r^2 + m^2 y^2 + n^2 z^2$ made by the plane $a x + b y + c z = 0$.

This is again an optimization problem involving a surface and a plane. The process involves using Lagrange multipliers, with the surface equation serving as the objective function and the plane equation as the constraint.

Problem 20: Find the lengths of the axes of the section of the ellipsoid $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$ by the plane $l x + m y + n z = 0$.

This problem asks for the lengths of the axes of the section formed by intersecting the ellipsoid with a plane. The general approach involves parameterizing the intersection and determining the principal axes using geometric methods.

Problem 21: Find the maximum and minimum radii vectors of the section of the surface $a x^2 + b y^2 + c z^2 = 1$ by the plane $l x + m y + n z = 0$.

Similar to problems 18 and 19, this requires optimizing the function $a x^2 + b y^2 + c z^2 = 1$ under the constraint $l x + m y + n z = 0$, and again using Lagrange multipliers.

Would you like me to proceed with detailed steps for each of these problems, focusing on the setup and solution using Lagrange multipliers, or summarize the main ideas? Let me know!