91Ó°ÊÓ

In mathematics, a vector function, especially in the context of parametric surfaces, is a brilliant tool to describe surfaces in 3D space. A vector function typically depends on two parameters, often denoted as \( u \) and \( v \), which can vary over a range of values. The given surface is described using the vector function \( \mathbf{r}(u, v) = \langle u+v, u-2v, 3+u-v \rangle \). This representation allows us to parameterize the surface, which is like creating a map that assigns each point on the surface certain parameter values.
By using vector functions, complex surfaces that cannot be expressed using a single equation become more accessible and easier to analyze. In this context, each component of the vector function corresponds to a coordinate in 3D space:

• \( u+v \) determines the \( x \)-coordinate.
• \( u-2v \) determines the \( y \)-coordinate.
• \( 3+u-v \) determines the \( z \)-coordinate.

This parameterization method simplifies working with 3D surfaces, allowing us to find whether certain points lie on the described surface by determining suitable \( u \) and \( v \) values.

When verifying whether a point, such as \( P(4, -5, 1) \), lies on the surface defined by a vector function, we substitute these coordinates into the vector function and solve for the parameters, \( u \) and \( v \). Solving these equations systematically can confirm the point's presence on the surface.
Taking example point \( P \): the coordinates satisfy the vector function equations \( u+v = 4 \), \( u-2v = -5 \), and \( 3+u-v = 1 \). Solving these equations step-by-step involves simultaneously solving linear equations. Here's how it's done:

• Combined equations such as \( u-v = -2 \) and \( u+v = 4 \) help simplify and find common solutions.
• Add and subtract the equations to isolate \( u \) or \( v \). For instance, from adding \( (u-v) + (u+v) = -2 + 4 \), we find \( 2u = 2 \), hence \( u = 1 \).
• Substitute back to find \( v \): \( u+v = 4 \) gives \( v = 3 \).

Through step-by-step solving, we determine successful parameter values, closely connecting algebra with geometric interpretation.

Understanding points in three-dimensional space is essential for working with parametric surfaces and vector functions. Each point is represented by a triplet \( (x, y, z) \), which corresponds to its location along the \( x \), \( y \), and \( z \) axes, respectively. This is crucial for understanding whether points like \( P(4, -5, 1) \) and \( Q(0, 4, 6) \) belong to the surface.
With a parametric surface, as structured by our vector function, \( (x, y, z) \) coordinates are found for every choice of parameters \( u \) and \( v \). Practically, this means that:

• The \( x \)-coordinate equals \( u + v \).
• The \( y \)-coordinate equals \( u - 2v \).
• The \( z \)-coordinate equals \( 3 + u - v \).

This system allows us to navigate between the parameterized form of the surface and real-world geometric coordinates. When working with 3D coordinates, verifying that a certain point lies on a surface involves matching these calculated coordinates to the given point's coordinates. If the equations don’t reach a match, as with point \( Q(0, 4, 6) \), the point does not lie on the surface.