Equation of the tangent line in the direction of a vector

[alex steve]

I am having issues figuring out how to do the "in the direction of the vector" part of my problem

I have found the equation of the tangent line but i do not know how to the the next part.

My question asks:

Find the equation of the tangent line to the surface defined by the function f(x,y) = x + e^(xy) at point (2,-1) in the direction of the vector u = <1,-2>

Would i have to figure out the gradient vector of my equation that i find?

 

[andrewkirk]

The phrasing of the question is a little obscure, because the vector u is a 2D vector in a plane that is not tangent to the surface.
I think what they mean is that they want the equation of the tangent line to the surface whose projection on the x-y plane is the vector u. Or, what amounts to the same thing, the projection of u on the tangent plane to the surface at the given point.

One way to do that is to find the curved line that is the intersection between the surface and the vertical plane with equation y=-2x. Then find the tangent to that line at the given point.

There may well be a quicker way, but that's all that occurs to me right now.