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Basically, I'm hardcore studying for my Calc III exam (the final is in a few weeks) and I came across an interesting lapse in my understanding (well many in fact, but one in particular).

First of all I can assume that a tangent and a normal/perpendicular vector are NOT the same thing, yet for some reason they both have pretty much the same characteristics.

I know that finding at least one of them involves finding the partial derivatives of a given function, which then are used as the components of the gradient function (the upside down triangle thing) Right?

And a gradient gives you... the normal vector right? or something regarding the tangent plane at a point, whose normal vector is that of the surface of the function? I could be mistaken but my major malfunction seems to be that I can't differentiate (no math puns please- I'm doing more studying than sleeping) between finding the normal vector and the unit tangent vector, which in practice seem to be the same thing (at least for me.)

(It is at this point where I actually noticed the warning at the top of the page regarding the template I am supposed to use. I apologize, and if a problem arises I'll be glad to repost using the template, which at this point seems pointless...)