I'm working through some problems asking to find the equation of a tangent plane and the normal line to the surface. I notice that some example questions parameterize the curve before solving and give an answer in terms of the parametric equations. Some give an equation of the tangent plane and then only use parametric equations for the normal line.
My understanding of parameterization is that it allows us to represent the same curve from a different perspective and describes the same curve with respect to different parameters, and that the purpose of this is often to make the the equation easier to work with under certain conditions.
However, with these types of problems, how do I know when it is useful to parameterize? Why do some solutions only write the equations of the normal lines with respect to the parameter t after finding the equation of the tangent plane with respect to x and y?
A line in three or more dimensions generally can’t be implicitly represented by a single equation. You either need a system of implicit equations or a parameterization of the line. A tangent hyperplane, on the other hand, can be represented by a single implicit Cartesian equation, but it can also be parameterized. Ultimately, the choice or representation depends on what you want to do with that tangent or normal once you’ve found it.