91Ó°ÊÓ

In the context of parametric surfaces, the **surface equation** plays a crucial role. A surface is often described by a parametric vector function, denoted as \( \mathbf{r}(u, v) \). This function expresses the coordinates \( (x, y, z) \) of any point on the surface in terms of two parameters, \( u \) and \( v \). In this exercise, the surface equation is given as \( \mathbf{r}(u, v) = \langle 2u + 3v, 1 + 5u - v, 2 + u + v \rangle \).

Understanding this means that each point on the surface can be obtained by varying \( u \) and \( v \). The equations extracted from this description are:

• \( x = 2u + 3v \)
• \( y = 1 + 5u - v \)
• \( z = 2 + u + v \)

These equations allow us to test whether a given point lies on the surface by substituting the point's coordinates \( (x, y, z) \) into them to find corresponding values of \( u \) and \( v \). This is a typical problem in exploring the properties of surfaces in three-dimensional space.

The heart of determining whether points lie on such a surface involves solving **simultaneous equations**. These are sets of equations that must all be satisfied for the given variables. For our parametric surface problem, we set up a system of three equations derived from substituting the point coordinates into the parametric equations.

To check if a point is on the surface, we simultaneously solve:

• \( x = 2u + 3v \)
• \( y = 1 + 5u - v \)
• \( z = 2 + u + v \)

This means for each point, we substitute its \( (x, y, z) \) values into the equations and solve for \( u \) and \( v \).

The solution process involves:

• Using algebraic manipulation to solve for one variable in terms of the other from one of the equations.
• Substituting back into the other equations to find consistent values of \( u \) and \( v \).

This method ensures that both variables satisfy all equations simultaneously. Successfully solving them confirms that the point is indeed on the surface.

**Point verification** in parametric surfaces is all about determining if specific coordinates belong to the described surface. Once we manipulate and solve the simultaneous equations, we must verify if the solutions for \( u \) and \( v \) are consistent across all original parametric equations.

For point \( P(7, 10, 4) \), though calculations yield \( v = 3 \) and \( u = -1 \), verifying these in the equations proves inconsistent—showing the point doesn’t lie on the surface.

In contrast, for point \( Q(5, 22, 5) \), the values found, \( v = -1 \) and \( u = 4 \), satisfy all equations. Inserting these into all three of the original parametric equations holds true, confirming the point’s presence on the surface.

This verification stage is crucial as mistakes in earlier calculations could lead to incorrect conclusions. It’s always wise to ensure the values of \( u \) and \( v \) work in reality through such a comprehensive check.