Originally Posted by
MrCryptoPrime Question: Find an equation of the tangent plane to the parametric surface x=1rcosθ, y=4rsinθ, z=r at the point (sqrt(2),4sqrt(2),2) when r=2, θ=pi/4.
My attempt:
Let m = <rcosθ, 4rsinθ, r>, then
m_{r } = <cosθ, 4sinθ, 1>
m_θ = <-rsinθ, 4rcosθ, 0>
Normal = (m_r x m_θ) = [4sinθ*0-4rcosθ*1]i - [cosθ*0 - (-rsinθ*1)]j + [4rcosθ*cosθ - 4sinθ*-rsinθ]k = <-4rcosθ, -rsinθ, 4r> <--- here is your mistake
N at the point (sqrt(2),4sqrt(2),2) => <-4sqrt(2), sqrt(2), 8>
Now, <(x-sqrt(2), y-4sqrt(2), z-2> dot <-4sqrt(2), sqrt(2), 8> = 0
-4sqrt(2)((x-sqrt(2)) + (sqrt(2)(y-4sqrt(2))) + 8(z-2) = 0
z = (4sqrt(2)x - sqrt(2)y + 16) / 8
This is the wrong answer, could anyone point out where I went wrong or a better way to do this? Thanks!