visualizing vector calculus and analytic geometry

Hi,

I'm a public schools math teacher (in Germany) and I'll be teaching a
unit on analytical geometry in my 12 grade math course.

What I like to be able to do is to simply and quickly manipulate and
visualize vectors and planes and lines in R^3 using different
representations (parametric, vector, etc.), e.g. enter vectors and have
them appear in R^3, then e.g. multiply the vectors with a scalar, and
see them get longer, subtract two vectors and see the difference vector,
visualize parameter and vector rep. of planes and lines and their
intersection etc., not just plotting 3d surfaces of f(x,y).

I've tried the plotting in maxima, and I can't get it to plot single
points or vectors, just surfaces of f(x,y). The plotting in maxima
requires entering lots of parameters and I've not even found a way to
plot vectors, let alone easily manipulate them and  see the changes
quickly (I want to do this live in class, e.g. quickly entering the
vector and seeing it in R^3). I've also found no way to enter vectors,
or vector representations of lines and planes.

Can maxima do this? How?! If not, does anybody know of a usable free
software which can manipulate and visualize vectors, planes, etc. in 3d?


I'm really grateful for any suggestions! :)

Thanks

-j