Gradient along or normal to tangent hyperplane?

[cunning tendon]

Hello, I've read this: https://math.mit.edu/~djk/18_022/chapter03/section03.html
Specifically the part: "The gradient vector points normal to the tangent plane of f in two dimensions, and normal to the tangent hyperplane in higher dimensions."

I thought the gradient vector is ON the tangent hyperplane to the function (at some point) not normal to it? So in real valued functions the derivative represents the slope $m$ of a line on the tangent line and $-1/m$ is the normal line, not the other way around which is what this statement says. Unless I'm misunderstanding it, is it wrong?

[proven ridge]

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[ionic plume]

@cunning tendon you here ?

anyway, the thing to note here, is that if you have say a 2var function f(x, y), a normal vector (or a tangent vector) to the surface z=f(x,y) lives in 3 dimensional space

but the gradient only lives in "2D space", for any input (x,y), it will give you the change in input that increases f the fastest : it lives in the xy-plane in a sense

so saying the gradient is the normal vector, or a tangent vector to the surface is nonsense

however what is true is that when you know the gradient, you can compute easily a normal vector to the surface (see https://en.wikipedia.org/wiki/Normal_(geometry))

and if you project that normal vector onto the xy-plane

it will be in the same direction as the gradient

[cunning tendon]

this is get

the gradient of R^2 -> R is a 2x1 vector

[cunning tendon]

this i do not understand, you said the gradient gives you the direction of steepest ascend which i know but then if you project the normal vector of a surface on that surface, it looks like it points in the complete opposite direction

this is what i mean

[ionic plume]

if you project the normal vector of a surface on that surface
I talked about projecting the normal vector on the xy-plane, i.e. if you have a vector (x, y, z), you look at (x, y, 0)

it looks like it points in the complete opposite direction
when I say direction, I just mean lay on the same line pretty much

those two vectors are in the same direction, they have different orientations

[cunning tendon]

they have the same span but opposite directions

[ionic plume]

hmm

[cunning tendon]

ok, i get it, thanks for clearing it up