Perpendicular line to the Surface

τheory

1. The problem statement, all variables and given/known data
Consider the line perpendicular to the surface [itex]z=x^2+y^2[/itex] 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:

[itex]0=x^2+y^2-z[/itex]
[itex]∇F(x,y,z)= <2x,2y,-1>[/itex]
[itex]∇F(-1,2,0)= <-2,4,-1>[/itex]

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

[itex]L(t) = P + t∇F[/itex]
[itex]L(t) = <-1,2,0> + t<-2,4,-1>[/itex]

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

LCKurtz

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

τheory

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