- #1
fishturtle1
- 394
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Homework Statement
Use gradients to find an equation of the tangent plane to the ellipsoid ##\frac {x^2}{4} + \frac {y^2}{9} + \frac {z^2}{25} = 3## at ##P = (2, -3, -5)##.
Homework Equations
##\triangledown f## is a normal vector of f.
The Attempt at a Solution
Let ##w = \frac {x^2}{4} + \frac {y^2}{9} + \frac {z^2}{25}## be the level curve of f at w.
then ##w(2, -3, -5) = \frac {4}{4} + \frac {9}{9} + \frac {25}{25} = 3##.
So P is on the level curve w = 3. I can't figure out why we have to do this, I"m just doing it to follow the example. Isn't this arbitrary?
After that step, we solve for ##\triangledown f(P)##.
##\triangledown f(P) = <\frac x2, \frac {2y}{9}, \frac {2z}{25} >##
##\triangledown f(2, -3, -5) = <1, -\frac {2}{3}, -\frac {2}{5}>##
So the equation of the tangent plane is
##<1, -\frac {2}{3}, -\frac {2}{5}> \cdot <x - 2, y + 3, z + 5> = 0##
##(x - 2) + -\frac {2}{3}(y + 3) + -\frac {2}{5}(z + 5) = 0##
I'm even more confused to find the normal line at this P. Would it just be ##\vec {OP} + \triangledown f(P)## since ##\triangledown f(P)## is a normal vector to f?
I'm having A LOT of trouble visualizing what is going on here.. please help, thank you.