Eq. of tangent plane at a point.

[catch22]

Homework Statement

Homework Equations

The Attempt at a Solution

what is the difference between the form

z - z_o = f_x(x-x_o) + f_y(y-y_o)

and:

F_x(x_o,y_o,z_o) + F_y(x_o,y_o,z_o) + F_z(x_o,y_o,z_o)=0
____________________________________________________________________________using the first eq :

while the latter gives:

Am I doing something wrong? shouldn't they give the same answer?

 

[Student100]

Homework Statement

View attachment 91052

Homework Equations

The Attempt at a Solution

what is the difference between the form

z - z_o = f_x(x-x_o) + f_y(y-y_o)

and:

F_x(x_o,y_o,z_o) + F_y(x_o,y_o,z_o) + F_z(x_o,y_o,z_o)=0
____________________________________________________________________________using the first eq :
View attachment 91053

while the latter gives:
View attachment 91054

Am I doing something wrong? shouldn't they give the same answer?

You're doing something wrong, when you solve for $z-z_0$, what happens to the term in front of $(z-z_0)$, and what about your signs?

 

[catch22]

You're doing something wrong, when you solve for $z-z_0$, what happens to the term in front of $(z-z_0)$, and what about your signs?

are you talking about the first equation z - zo = fx(x-xo) + fy(y-yo)? I thought there aren't suppose to be any terms in front of (z−z0)

actually, in order to get z - zo = fx(x-xo) + fy(y-yo), didn't we have to divide everything by z so we had something like A/C = F_x and B/C = F_y

 

[catch22]

I think I found the problem.

if I use gradient, the F_z has to equate to -1, right?

 

[catch22]

You're doing something wrong, when you solve for $z-z_0$, what happens to the term in front of $(z-z_0)$, and what about your signs?

is this correct?

 

[HallsofIvy]

The difference between "z= f(x, y)" and "F(x, y, z)= 0" is that you are taking F(x, y, z)= f(x, y)- z. Taking the gradient \nabla F= <f_x, f_y, -1>[.

It is correct that F(1, -1, 4\sqrt{2})= \frac{1}{4}+ \frac{1}{4}+ \frac{32}{64}= 1 but that does NOT say "z= 1". You are told that at this point z= 4\sqrt{2}. The fact that F(1, -1, 4\sqrt{2})= 1 just verifies that the given point is on the given surface.

The tangent plane is z- z_0= f_x(x_0, y_0)(x- x_0)+ f_y(x_0, y_0)(y- y_0) only applies if you are given z= f(x, y) which is NOT the case here. You are given, rather, F(x, y, z)= x^2/4+ y^2/4+ z^2/64= 1. In order to convert that to the "z= f(x,y)" form you need to solve for z: z^2/64= 1- x^2/4- y^2/4, z/8= \sqrt{1- x^2/4- y^2/4}, z= 8\sqrt{1- x^2/4- y^2/4}.

Much simpler is to use the fact that if F(x, y, z)= 1 (or any constant) then \nabla F is perpendicular to the surface at any point. Since for any point (x, y, z) on the tangent plane to the surface at (x_0, y_0, z_0), the vector (x- x_0)\vec{i}+ (y- y_0)\vec{j}+ (z- z_0)\vec{k} is in the tangent plane, we must have \nabla F\cdot [(x- x_0)\vec{i}+ (y- y_0)\vec{j}+ (z- z_0)\vec{k}]= 0 which is the same as F_x(x_0, y_0, z_0)(x- x_0)+ F_y(x_0, y_0, z_0)(y- y_0)+ F_z(x_0, y_0, z_0)(z- z_0)= 0 as the equation of the tangent plane.

 

[catch22]

The difference between "z= f(x, y)" and "F(x, y, z)= 0" is that you are taking F(x, y, z)= f(x, y)- z. Taking the gradient \nabla F= <f_x, f_y, -1>[.

It is correct that F(1, -1, 4\sqrt{2})= \frac{1}{4}+ \frac{1}{4}+ \frac{32}{64}= 1 but that does NOT say "z= 1". You are told that at this point z= 4\sqrt{2}. The fact that F(1, -1, 4\sqrt{2})= 1 just verifies that the given point is on the given surface.

The tangent plane is z- z_0= f_x(x_0, y_0)(x- x_0)+ f_y(x_0, y_0)(y- y_0) only applies if you are given z= f(x, y) which is NOT the case here. You are given, rather, F(x, y, z)= x^2/4+ y^2/4+ z^2/64= 1. In order to convert that to the "z= f(x,y)" form you need to solve for z: z^2/64= 1- x^2/4- y^2/4, z/8= \sqrt{1- x^2/4- y^2/4}, z= 8\sqrt{1- x^2/4- y^2/4}.

Much simpler is to use the fact that if F(x, y, z)= 1 (or any constant) then \nabla F is perpendicular to the surface at any point. Since for any point (x, y, z) on the tangent plane to the surface at (x_0, y_0, z_0), the vector (x- x_0)\vec{i}+ (y- y_0)\vec{j}+ (z- z_0)\vec{k} is in the tangent plane, we must have \nabla F\cdot [(x- x_0)\vec{i}+ (y- y_0)\vec{j}+ (z- z_0)\vec{k}]= 0 which is the same as F_x(x_0, y_0, z_0)(x- x_0)+ F_y(x_0, y_0, z_0)(y- y_0)+ F_z(x_0, y_0, z_0)(z- z_0)= 0 as the equation of the tangent plane.

looks correct?

 

[HallsofIvy]

What are x_0, y_0 and z_0?

 

[catch22]

What are x_0, y_0 and z_0?

I used r⋅n = n ⋅ (1,-1,4sqrt(2))
to find my previous answer.

r being (x,y,z)

 

[HallsofIvy]

So you do not know what x_0, y_0, and z_0 are? Then you need to talk to your teacher.

 

[catch22]

So you do not know what x_0, y_0, and z_0 are? Then you need to talk to your teacher.

So you do not know what x_0, y_0, and z_0 are? Then you need to talk to your teacher.

they are the point given, (1,-1,4sqrt(2))

 

[HallsofIvy]

Yes. I asked that in post #8 and you did not answer. Now, my question is "why do you not have (x- 1), (y+ 1), and (z- 4\sqrt{2}) in your answer"?

 

[catch22]

that was what I initially had