Thank you for your quick answers.
I have to explain it a bit more. My first posting was written while I was sitting in my math lesson and I was in a hurry. I’d just like to apply the math knowledge in a sort of a blender addon or feature.
But first of all to the answers:
Well, that’s kind of the way I want to go. But I recently did a videotutorial on the 3D function surfaces, so I know, that this is not the desired effect, since it doesn’t work with vectors, but with implicit and explicit equations. In other words, I want to have full control over my model like in this addon with textfields and slide controls, yes. But I want to input other formated information. My main intention is to create a feature for educational purpose, to teach students how vectors and 3D math works - what would be better than a highly-sophisticated 3D Software?! 
That’d be nice, but I think that’s too complicated for my intention. Since there are only lines and planes to be shown, curves are not really needed. And if yes, they should be created with vector-based methods.
Exactly this is what I’m talking about
Normally there is a problem with planes - they go on into infinity. So I need a way to “cut” them off at certain limitations. I thought about a slider control to set the interval of all the possible values for the parameters respectively.
So let’s start from the beginning.
Example 1: Parametric line equation
These start with a vector that represents the origin. So if the origin vector is e.g. (2|3|-5), then we “walk” 2 units in x-direction, 3 units in y-direction and -5 units in z-direction, measured from the coordinate space origin. This is part one of our equation.
Part two says, where the line should expand, namely in what direction. So we need a second vector. This is generelly a parameter (e.g. μ) multiplied by a vector, e.g. (-3|2|2). Since μ can be every possible number, we can set points all along from our origin point in our defined direction and (with negative values) even in the opposite direction. But that’s also our problem (the line would go into infinity - this could cause problems in rendering), which can be solved by defining a limitation for μ.
I did some image manipulation on how slide controls could look like:

Of course, changes of the values should be dynamic and in realtime!
Example 2: Parametric plane equation
This should work just like the line settings, but with two directional vectors, instead of one(with the scaling parameters e.g. λ and μ). So eventually the plane is described by three vectors:
- a vector that says where the origin of the plane is
- the first directional vector
- and the second directional vector which span the plane.
And that concludes the technical part. Hopefully my english is not so bad, that you don’t understand anything - if so, then just tell me! I’ll try to explain it more clearly.
Addition: Parameters in vector coordinates
Something I ain’t clear myself about is the use of more parameters. For example for the coordinate values. So, like a vector in this manner (1-a|a+5|a). This would mean, that there should appear a new slidecontrol for every new parameter or there could be a fixed number of parameters in the beginning. I think, the effect it would have, would be pretty impressive!
Summary:
Simply I need a script that is able to update the mesh dynamically. The user should be able to set his points and vectors with text input fields as well as slide controls. The logic in python would then calculate the position of points with simple analytic geometry methods. Hopefully it is easy to display points more visible in the 3D view, by creating spheres or textfields at the resultant coordinates or even display the vector with an actual vector mesh (but that’s just for visualizing). Extensions then could be to calculate intersections between planes and lines, but that should be easily possible when I know how to write the code for the original problem, since this is only a bit more math.
Hmm, this shouldn’t be no rocket science, should it?! :-/
What do you think? How would you rate my idea for this project(I can cope with critics)?
Thank you all for contributing ideas!