Finding Parallel Tangent Planes on a Surface

[EngnrMatt]

Homework Statement

Find the points on the surface \(4 x^2 + 2 y^2 + 4 z^2 = 1\) at which the tangent plane is parallel to the plane \(4 x - 3 y - 2 z = 0\).

Homework Equations

Not sure

The Attempt at a Solution

What I did was take the gradient of both functions, the surface function as f, and the plane as g, so I got ∇f = <8x,4y,8z> and ∇g = <4,-3,-2>. Knowing that these gradients are normal vectors, and the planes would be in parallel, therefore having the same normal vectors, I set the components equal to each other and got x = 1/2, y = -3/4, z = -1/4. However, these are not the points, and I have rune out of ideas on how to work the problem.

 

[ehild]

Homework Statement

Find the points on the surface \(4 x^2 + 2 y^2 + 4 z^2 = 1\) at which the tangent plane is parallel to the plane \(4 x - 3 y - 2 z = 0\).

Homework Equations

Not sure

The Attempt at a Solution

What I did was take the gradient of both functions, the surface function as f, and the plane as g, so I got ∇f = <8x,4y,8z> and ∇g = <4,-3,-2>. Knowing that these gradients are normal vectors, and the planes would be in parallel, therefore having the same normal vectors, I set the components equal to each other and got x = 1/2, y = -3/4, z = -1/4. However, these are not the points, and I have rune out of ideas on how to work the problem.

The normal vectors should be parallel, not equal. <8x,4y,8z>= K<4,-3,-2>, K is a constant.


ehild

 

[EngnrMatt]

Alright, I got it. Thank you very much!