Tangent plane approximation via parametric equations

Hi everyone,

I'm an 18-year-old from Germany and I'm making use of MIT's OpenCourseWare programme. Currently, I'm watching the Calculus II course, and am having some trouble understanding how to find the equation:
z=z0 + a(x-x0) + b(y-y0) by using parametric equations/vectors.
a and b being the partial derivatives of x and y. Now, while I understand the formula itself, the lecturer said that it can also be obtained by first finding the parametric equations for the tangents, then finding vectors along them and finally finding the plane they are on by calculating the cross-product. That seems straightforward enough, but somehow I can't seem to get it right.
I thought I'd use the formula Q0+ t(Q0Q1) to find the parametric equation. So, for the x-line, I thought I could define Q0 as (x0|y0|z0) and Q1 as (x|y0|a(x-x0)+z0). Then Q0Q1 would be <x-x0, 0, a(x-x0)> and Q0Q2 (for y) <0, y-y0 ,a(y-y0)>. The cross-product, however, is rather strange and somehow I think I'm going about this the wrong way...
EDIT: I've added a PDF to make it a little clearer...

Thanks a lot for your efforts,

Martin
 

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