# Calculus Question: Gradients & Normals

1. Oct 19, 2004

### Lomion

Geometry Question: Gradients & Normal lines & planes

This is a question from a past midterm that I'd appreciate some help with. It deals with the gradient as a normal. I'm not having trouble actually obtaining the gradient, but I am having trouble with some of the geometry involved, so any help would be appreciated!

Consider the function:

$$y = \sqrt{x^2 + z^2}$$

Give the equation for 2 planes whose intersection is the normal line to this surface at $$(1,4,\sqrt{15})$$.

I found the value: $$\nabla f(1,4,\sqrt{15}) = (1/4, -1, \sqrt{15}/4)$$.

And the equation for the normal line is:

$$r(t) = (1,4,\sqrt{15}) + t(1/4, -1, \sqrt{15}/4)$$

My question is: How do I find two planes that intersect in this line?

I think I should parametrize the variables, so

x = 1 + 1/4t
y = 4 - t
$$z = \sqrt{15} + \sqrt{15}/4 * t$$

But that's where I get lost. Can someone just point me in the right direction in terms of what equations to set up?

Thanks!

Last edited: Oct 19, 2004