Expressing a circle in the form r(t) = x(t)i + y(t)i +z(t)k

In summary, Laurence has learned about grad, div, and curl, but is struggling with a question on expressing a circle in a specific form. They are looking for someone to take them through the steps and provide a general equation. They also mention the possibility of the circle being in other planes. They thank the person who provided them with a solution for the specific case and state that they don't need anything more complicated.
  • #1
Lorentz_F
5
0
I learned all the hard (at my level) bits like grad, div, curl and now I'm falling at the first hurdle on the exam papers:

The question is: Express the circle of radius 3, centred at (1,0,3) and lying in the (x,z) plane in the form of r(t) = x(t)i + y(t)j + z(t)k

I was hoping there was someone who wouldn't mind taking me through this step by step especially if there is a simple general equation as I have looked everywhere. And also what to do if it is in the other planes etc.

Thank you,

Laurence

P.s. Sorry if this qualifies as homework, I thought that it didn't.
 
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  • #2
y(t)=0
x(t)=3(cos(t) - 1)
z(t)=3(sin(t) - 3)

When the circle is in a plane perpendicular to one of the axes, it is easy as you can see.

For the general case, it is more complicated. You can set it up in a coordinate system in the given plane and then transform into the global coordinates.
 
  • #3
:approve: Thanks alot, I don't think i'll be needing anything more complicated than this so greatly appreciated.

Laurence
 

FAQ: Expressing a circle in the form r(t) = x(t)i + y(t)i +z(t)k

1. What is the purpose of expressing a circle in the form r(t) = x(t)i + y(t)i +z(t)k?

The purpose of expressing a circle in this form is to use parametric equations to describe the position of a point on the circle at any given time. This form is useful for studying the motion of the circle, such as its velocity and acceleration, and for solving problems involving circles in physics and engineering.

2. How is r(t) = x(t)i + y(t)i +z(t)k different from the traditional equation of a circle, x^2 + y^2 = r^2?

The traditional equation of a circle, x^2 + y^2 = r^2, describes a circle in Cartesian coordinates, where the position of a point is given in terms of its x and y coordinates. On the other hand, r(t) = x(t)i + y(t)i +z(t)k uses parametric equations, where the position of a point is described in terms of a parameter t. This allows for more flexibility in studying the motion of the circle.

3. Can r(t) = x(t)i + y(t)i +z(t)k be used to express all types of circles?

Yes, r(t) = x(t)i + y(t)i +z(t)k can be used to express all types of circles, including circles with different radii and centers. By varying the values of x(t), y(t), and z(t), the equation can describe circles of any size and orientation in three-dimensional space.

4. How is the position of a point on a circle determined using r(t) = x(t)i + y(t)i +z(t)k?

The parameter t represents time, and plugging in different values for t will give the position of a point on the circle at different times. For example, if t = 0, the equation will give the initial position of the point on the circle, and as t increases, the equation will give the position of the point at later times.

5. Are there any limitations to expressing a circle in the form r(t) = x(t)i + y(t)i +z(t)k?

One limitation of this form is that it can only be used to describe circles in three-dimensional space. It cannot be used for circles in higher dimensions. Additionally, the parameter t is typically limited to positive values, so negative time values cannot be used to describe the circle's motion.

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