How can the length of a normal vector matter?

[ainster31]

Homework Statement

Homework Equations

The Attempt at a Solution

Solution:

Graph:

For part a, I understand mathematically why the value of c matters. What I don't understand is how it can possibly matter intuitively.

I get that $\overrightarrow { \nabla } F(x_{ 0 },y_{ 0 },z_{ 0 })=(0,c,0)$ and therefore, c can't be any value. My question is: how can a tangent plane possibly depend on the length of the normal vector? If the vector is normal, then the length shouldn't matter because the vector will always be normal for all values for c except 0.

 

[vela]

You need two pieces of information to specify the plane: the normal and a point on the plane. $c$ goes into figuring out what point the plane passes through.

 

[ainster31]

You need two pieces of information to specify the plane: the normal and a point on the plane. $c$ goes into figuring out what point the plane passes through.

What are the effects on a tangent plane when you scale up and down ∇F but keep ∇F in the same direction?

 

[vela]

$\nabla F$ is determined by F, so you don't have the freedom to arbitrarily rescale it. If you mean what happens if you say, for example, $\nabla F = (0, c/2, 0)$ instead of $\nabla F = (0, c, 0)$, why don't you try it and see what happens?