Tangent Space

Hi,

I have been asked to answer this question...

Problem 3. A surface M ⊂ R3
is given by the equation z = x^2 + y^2

Check that the vector u = (3, −1, 4) ∈ R3 belongs to the tangent space to M at the point
x = (1, 1, 2) ∈ M and expand u over the basis ex, ey.

The second part is no problem, I can expand it over the basis given that u is in the tangent space - but I must be missing something obvious because I can't seem to establish why u is in the tangent space!

Thanks for your help

I'm not an expert but would it not be sufficient to show that $\nabla f(\mathbf{x}) \cdot \mathbf{u}=0$, where $f$ is the equation of the surface, since the gradient is normal to the surface, and hence normal to the tangent plane.
You may have better luck posting here.

Thanks for your reply. That makes sense, and I think would be sufficient