I have question: Find the equation for the tangent plane and the normal line of the surface
$f(x,y,z)=x^2+y^2+z$
at point $(1,1,1)$
For the tangent plane I got,
$z=2x+2y+z-2$
is this correct, if so from here how do I find the normal line?
Asked
Active
Viewed 828 times
2
maponda1
- 83
-
Are you missing something? In the title you say $f(x,y,z) = x^2+y^2 +z$ and in the question $f(x,y,z) = x^2+y^2+7$. Which one is right? – Mark Fantini Jun 01 '14 at 20:48
-
its $z$ not $7$, thanx for that observation, twas typing error – maponda1 Jun 01 '14 at 21:09
1 Answers
0
The normal vector can be found using the gradient of the function computed at the point. Therefore $$\nabla f (1,1,1) = (2x,2y,1) \bigg\vert_{(x,y,z)=(1,1,1)} = (2,2,1)$$ is the normal vector of the tangent plane at $(1,1,1)$. Writing it as $2x+2y+z=d$ we substitute $(1,1,1)$ into the equation to find $d$, obtaining $$2x+2y+z=5.$$ As for the normal line, write it as every line: $$r: P_0 + t D,$$ where $P_0$ is the starting point and $D$ is the directing vector, the first vector we found..
Mark Fantini
- 5,523