What is the angle between the normals to the surface at two given points?

[msslowlearner]

Homework Statement

find the angle between the normals to the surface xy=z² at the points (1,4,2) and (-3,-3,3)

Homework Equations

none

The Attempt at a Solution

del S = 2zy i + 2zx j
and at the two points, del S = 16i+4j and del S = -18i-18j

using the dot product, i got cos theta = -5/(4*sqrt 34)

which is NOT the answer. Is my partial derivative equation right ? Please help.

 

[LCKurtz]

Homework Statement

find the angle between the normals to the surface xy=z² at the points (1,4,2) and (-3,-3,3)

Homework Equations

none

The Attempt at a Solution

del S = 2zy i + 2zx j
and at the two points, del S = 16i+4j and del S = -18i-18j

using the dot product, i got cos theta = -5/(4*sqrt 34)

which is NOT the answer. Is my partial derivative equation right ? Please help.

No, it isn't right. Write f = xy - z² = 0 and calculate

∇f = <f_x,f_y,f_z>

 

[Mark44]

Homework Statement

find the angle between the normals to the surface xy=z² at the points (1,4,2) and (-3,-3,3)

Homework Equations

none

The Attempt at a Solution

del S = 2zy i + 2zx j
and at the two points, del S = 16i+4j and del S = -18i-18j

using the dot product, i got cos theta = -5/(4*sqrt 34)

which is NOT the answer. Is my partial derivative equation right ? Please help.

It doesn't look right to me. I don't understand how you got what you did.

The equation for your surface is xy = z², or equivalently, xy - z² = 0. The implied function here is F(x, y, z) = xy - z². Your surface is the set of points (x, y, z) for which F(x, y, z) = 0.

For the gradient, calculate [itex]\nabla F[/itex] and evaluate it at the two points. Each will give you a vector. From these vectors you can determine the angle, using the approach that you showed above.

 

[msslowlearner]

sorry .. should have checked the partial derivative part. I thought i got the idea wrong. Its heartening to know I'm on the right track atleast :) thankyou

 

[SammyS]

stallionx,

Please don't post complete solutions.

See the rules for posting Homework Help:

"On helping with questions: Any and all assistance given to homework assignments or textbook style exercises should be given only after the questioner has shown some effort in solving the problem. If no attempt is made then the questioner should be asked to provide one before any assistance is given. Under no circumstances should complete solutions be provided to a questioner, whether or not an attempt has been made."
​

It may still be possible to Edit your post.

 

[stallionx]

stallionx,

Please don't post complete solutions.

See the rules for posting Homework Help:

"On helping with questions: Any and all assistance given to homework assignments or textbook style exercises should be given only after the questioner has shown some effort in solving the problem. If no attempt is made then the questioner should be asked to provide one before any assistance is given. Under no circumstances should complete solutions be provided to a questioner, whether or not an attempt has been made."
​

I am sorry, Sir.

 

[stallionx]

For a tangent plane equation to the z=f(x,y))

z-z0=(delz/delx)(x-x0)+(delz/dely)(y-y0)

which says the normal vector is < delz/delx , delz/dely , -1>

Find those two vectors for 2 different points

Dot product these vectors

V1 * V2 = length(V1)*length(V2) * Cos(theta)

Theta = ACOS ( V1*V2)/ ( product of lengths of perpendicular/ orthogonal vectors )

 

[stallionx]

For a tangent plane equation to the z=f(x,y))

z-z0=(delz/delx)(x-x0)+(delz/dely)(y-y0)

which says the normal vector is < delz/delx , delz/dely , -1>

Find those two vectors for 2 different points

Dot product these vectors

V1 * V2 = length(V1)*length(V2) * Cos(theta)

Theta = ACOS ( V1*V2)/ ( product of lengths of perpendicular/ orthogonal vectors )

DOT PRODUCT can be found from coordinates multiplication and addition

CASE : (x,y,z) dot (a,b,c) is xa+by+cz