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Parametric equations provide an alternative to expressing a surface or a curve through a single equation by instead using multiple equations to represent relations between the variables independently. In the context of surfaces, they allow us to describe a surface using two parameters, which can be thought of as coordinates along the surface itself. For instance, if we have the equation of a plane such as \(z = 3x + 4y\), we can describe every point on this plane by selecting two independent parameters, such as \(t\) for \(x\) and \(s\) for \(y\).

So, our parametric equations become \(x(t, s) = t\), \(y(t, s) = s\), and \(z(t, s) = 3t + 4s\). This representation is powerful because it allows us to traverse the surface by simply varying the parameters \(t\) and \(s\), which can provide a deeper understanding of the geometry of the surface.

Vector functions are mathematical expressions where each component function refers to a different dimension of space, and they output vectors rather than scalars. In a three-dimensional space, a vector function will generally have three component functions, one for each coordinate axis.

For our exercise, the vector function that represents the surface is \(\vec{r}(t, s) = \langle t, s, 3t + 4s \rangle\). This notation with the angle brackets \(\langle \cdot \rangle\) signifies a vector with the given component functions. As \(t\) and \(s\) change, the output of the vector function moves accordingly in space, tracing out the surface. This is a clear and convenient method for visualizing and working with surfaces in three-dimensional spaces.

Surface equations in their standard form often relate two or more variables in a single equation, representing a continuous surface in space. However, by using a parametric representation, we can define these surfaces with greater flexibility and express complex surfaces that might be difficult or impossible to represent with a single implicit equation.

In the context of the given problem, the standard form equation of the surface is \(z = 3x + 4y\), which is an equation of a plane. Once we choose parameters and establish the parametric equations, we effectively reframe this standard form into a format that provides a direct method for generating every point on the surface. Here, the parametric representation does not change the underlying surface; it just provides a different, often more practical, method of describing it.