# Perpendicular line to the Surface

1. May 13, 2012

### τheory

1. The problem statement, all variables and given/known data
Consider the line perpendicular to the surface $z=x^2+y^2$ at the point where x = −1 and y = 2 Find a vector parametric equation for this line in terms of the parameter t.

3. The attempt at a solution
I wasn't quite sure how to go about with this problem so I just went along with the following ideas. I first took the gradient of the function at that point:

$0=x^2+y^2-z$
$∇F(x,y,z)= <2x,2y,-1>$
$∇F(-1,2,0)= <-2,4,-1>$

Then I constructed the vector parametric equation of the line at that point:

$L(t) = P + t∇F$
$L(t) = <-1,2,0> + t<-2,4,-1>$

Afterwards, I submitted this equation, only finding that it was incorrect; can someone explain to me what went wrong here?

2. May 13, 2012

### LCKurtz

When $x=-1$ and $y=2$, $z$ isn't zero.

3. May 13, 2012

### τheory

Wow haha that was a horrible miscalculation on my part. Thanks for pointing that out!

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