OK, this is really confusing me. Mostly because i suck at spatial stuff. If the gradient vector at a given point points in the direction in which a function is increasing, then how can it be perpendicular to the tangent plane at that point? If it's perpendicular to the tangent plane, wouldn't it be perpendicular to the function too? This is the Wikipedia image for the gradient of a function, and it's pretty much what I imagine when I think of it: http://upload.wikimedia.org/wikipedia/en/3/31/Gradient99.png" [Broken] but if those lines were perpendicular to the tangent planes at their given points, wouldn't they all be pointing away from the graph (like "hairs")? Or is it just saying that it's perpendicular to the level curves? -- the pictures are very confusing and they always look like it's pointing away from a tangent plane, which makes no sense to me. EDIT: I think I got confused because it was an example with a function of three variables f(x,y,z) and I was thinking about it in f(x,y).