Translate the rectangular equation to spherical

In summary, Translate the rectangular equation to spherical and cylindrical equations. ρ^2+2ρsinϕsinθ-3ρsinϕcosθ=25. Then factored out a ρ and solved for ρ.
  • #1
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Translate the rectangular equation to spherical and cylindrical equations.

http://www.texify.com/img/%5CLARGE%5C%21x%5E2%2By%5E2%2B2y-3x%2Bz%5E2%3D25.gif [Broken]
 
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  • #2
whynot314 said:
Translate the rectangular equation to spherical and cylindrical equations.

http://www.texify.com/img/%5CLARGE%5C%21x%5E2%2By%5E2%2B2y-3x%2Bz%5E2%3D25.gif [Broken]
Hello whynot314. Welcome to PF!

What have you tried?

Where are you stuck?

We can't help you until you show us what you tried. It in the rules .
 
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  • #3
I first transformed x^2+y^2+z^2 into ρ^2

ρ^2+2ρsinϕsinθ-3ρsinϕcosθ=25

then factored out a ρ

ρ(ρ+2sinϕsinθ-3sinϕcosθ)=25

I want to solve for ρ,

I am confused as to, weather or not subtract 25 and set it equal to zero or keep it like this and have two equations where

ρ=25

and


ρ=25-2sinϕsinθ+3sinϕcosθ
 
  • #4
Then what you need is a good class in arithmetic! If ab= c it does NOT follow that "a= c or b= c!

If that isn't enough, if xy= 10, it does NOT follow that either x= 10 (and y= 1) or y= 10 (and x= 1). There are an infinite number of soutions to xy= 10.
 
  • #5
whynot314 said:
I first transformed x^2+y^2+z^2 into ρ^2

ρ^2+2ρsinϕsinθ-3ρsinϕcosθ=25

then factored out a ρ

ρ(ρ+2sinϕsinθ-3sinϕcosθ)=25

I want to solve for ρ,

I am confused as to, weather or not subtract 25 and set it equal to zero or keep it like this and have two equations where

ρ=25

and

ρ=25-2sinϕsinθ+3sinϕcosθ
How can you say ρ = 25 ?

ρ2 + 2ρsinϕsinθ-3ρsinϕcosθ = 25 is quadratic in ρ .

Complete the square or use the quadratic formula to solve for ρ.

For either, you may want to rewrite your equation as:
ρ2 + 2ρsinϕ(sinθ-cosθ) = 25​
To complete the square, add
[itex]\sin^2\phi\left(\sin\theta-\cos\theta\right)^2[/itex]​
to both sides.

To use the quadratic formula instead, subtract 25 from both sides of
ρ2 + 2ρsinϕ(sinθ-cosθ) = 25 .​
 

What is the purpose of translating a rectangular equation to spherical?

Translating a rectangular equation to spherical allows for easier visualization and understanding of three-dimensional geometric figures and equations. It also allows for the use of different coordinate systems to solve problems.

How do you convert a rectangular equation to spherical?

To convert a rectangular equation to spherical, you can use the following formulas:
x = r * sin(theta) * cos(phi)
y = r * sin(theta) * sin(phi)
z = r * cos(theta)
where r is the distance from the origin, theta is the angle between the positive z-axis and the line segment connecting the origin to the point, and phi is the angle between the positive x-axis and the projection of the line segment onto the xy-plane.

What is the relationship between rectangular and spherical coordinates?

Rectangular coordinates and spherical coordinates are two different ways of representing points in three-dimensional space. Rectangular coordinates use the x, y, and z axes, while spherical coordinates use the distance from the origin, and two angles, theta and phi, to specify a point. The two coordinate systems are related by the formulas mentioned in the previous question.

What are the advantages of using spherical coordinates over rectangular coordinates?

Spherical coordinates are advantageous for solving problems involving spherical shapes or objects, such as the Earth or planets. They also allow for easier visualization and understanding of symmetrical figures and equations. Additionally, some equations may be simpler and more compact in spherical coordinates.

What are some real-life applications of translating a rectangular equation to spherical?

Translating rectangular equations to spherical coordinates is commonly used in fields such as physics, astronomy, and engineering. It is used to solve problems involving the motion of objects in a spherical environment, such as orbiting satellites or planets, and to analyze the behavior of waves and other physical phenomena. It is also used in computer graphics to render 3D images and animations.

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