Difficult multivariable problem to find equation for 3d surface

In summary, we are trying to find the equation for a given surface containing all points G such that the distance from G to the plane z=4 is double the distance from point G to the point (2, -3, 1). By using the distance formula for a point to a plane, we can get an expression for the distance between the point and the plane, which is D1=|z-4|. The distance between the point G and (2, -3, 1) is D2=sqrt((x-2)^2 + (y+3)^2 + (z-1)^2 ). By setting 2D1=D2, we can solve for the equation of the surface, which is
  • #1
Jimmy5050
7
0

Homework Statement



A given surface contains all points G such that the distance from G to the plane z=4 is double the distance from point G to the pt. (2, -3, 1). Find eqn for the surface.

Homework Equations



I thought the distance formula for a point to a plane would help, but I can tell that the equation is going to have to be that of a parabola.

The Attempt at a Solution



I've made at least 5 unsuccessful attempts, and have NO idea where to go from here.
 
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  • #2
Say the point G has coordinate (x, y, z). Can you get expressions for the distance between this point and the plane z=4 and for the distance between this point and (2, -3, 1)?
 
  • #3
For the distance formula between arbitrary point (x,y,z) on the surface to the plane z=4 would be;

D1= abs(ax+by+cz+d)/sqrt(a^2+b^2+c^2) where n=<a,b,c> is the normal vector to the plane z=4, which we can say is <0,0,1>.

So D1=abs(0x+0x+1z)/sqrt(0^2+0^2+1^2) = abs(z)


And the distance from the point G (x,y,z) and the point (2,-3,1) is

D2=sqrt( (x-2)^2 + (y+3)^2 + (z-1)^2 )

So 2D1=D2

So 2z=sqrt( (x-2)^2 + (y+3)^2 + (z-1)^2 )


I feel like something is wrong however, THANKS for the help!
 
  • #4
Jimmy5050 said:
For the distance formula between arbitrary point (x,y,z) on the surface to the plane z=4 would be;

D1= abs(ax+by+cz+d)/sqrt(a^2+b^2+c^2) where n=<a,b,c> is the normal vector to the plane z=4, which we can say is <0,0,1>.

So D1=abs(0x+0x+1z)/sqrt(0^2+0^2+1^2) = abs(z)


And the distance from the point G (x,y,z) and the point (2,-3,1) is

D2=sqrt( (x-2)^2 + (y+3)^2 + (z-1)^2 )

So 2D1=D2

So 2z=sqrt( (x-2)^2 + (y+3)^2 + (z-1)^2 )


I feel like something is wrong however, THANKS for the help!
This is almost right, but you got the distance between the point and the plane z=4 wrong because you set d=0. The origin, for example, is a distance 4 away from the plane, but |z|=0. Similarly, the point (0,0,4) is on the plane, so there's 0 distance between it and the plane, not a distance of |z|=4. Can you see how to fix your answer?
 
  • #5
I'm still not understanding what I have to correct.

I guess I don't really understand what should be plugged in for d. From my understanding, d is going to have to be some type of function because it is changing as the surface changes to different points "G"?
 
  • #6
Your formula for D2 is correct. Your formula for D1, however, isn't. I gave you two examples where it clearly gives the wrong answer.

When I said d=0, I was referring to the d which appears here:
D1= abs(ax+by+cz+d)/sqrt(a^2+b^2+c^2)

In your next step, it was gone, so I assumed you set it to 0.
 
  • #7
Ok... Since the distance from the point to the origin is d, then we can say that the distance from the point to the plane would be z-4

So in the equation, d = z-4

Thanks for all the help!
 
  • #8
Is that d or what you called D1 earlier? Remember d is a constant in that formula; z shouldn't appear in it.
 
  • #9
I meant it as d, not D1.

I now realize its a constant, so solving through with that same logic should just give d= -4?
 
  • #10
Yes. If you rewrite the equation of the plane as z-4 = 0, so that it's in the form of ax+by+cz+d=0, the -4 is d, so D1=|z-4|.
 

Related to Difficult multivariable problem to find equation for 3d surface

What is a multivariable problem?

A multivariable problem is a mathematical problem that involves more than one independent variable or parameter. This means that there are multiple factors that can affect the outcome of the problem, making it more complex than a problem with only one variable.

Why is finding an equation for a 3D surface difficult?

Finding an equation for a 3D surface is difficult because it involves multiple variables and parameters. The surface may not be a simple geometric shape, making it challenging to find a single equation that accurately represents it. Additionally, there are often many possible equations that could describe the surface, making it difficult to determine which one is the most accurate.

What are some techniques for solving difficult multivariable problems?

Some techniques for solving difficult multivariable problems include using calculus, linear algebra, and computer simulations. These methods can help to break down the problem into smaller, more manageable parts and find a solution that takes into account all of the variables and parameters involved.

How can I visualize a 3D surface to better understand it?

One way to visualize a 3D surface is by creating a graph or plot of the surface. This can help you to see the shape and characteristics of the surface and better understand how the different variables and parameters affect it. There are also software programs and tools available that can create 3D models of surfaces for visualization.

What are some real-world applications of solving difficult multivariable problems?

Solving difficult multivariable problems has many real-world applications, such as in engineering, physics, economics, and biology. For example, in engineering, multivariable problems can be used to design and optimize complex systems, such as aircraft or car engines. In physics, multivariable problems are essential for understanding and predicting the behavior of complex systems, such as fluid dynamics or quantum mechanics.

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