Parametrization homework help

In summary: What kind of a shape do you thinkz^2=x^2+y^2is?There are more than one way to parameterize a curve, so your answer and the books answer needn't agree. [edit:] But I note an error. Since you've chosen x=cos(t), y=sin(t) then x^2+y^2=1 you've implicitly added then another constraint and you cannot satisfy z^2 = x^2+y^2. Rather try x = z\cdot\cos(t) and y=z\cdot\sin(t)
  • #1
Weave
143
0

Homework Statement


Find a vector function that represents the curve of the intersection of two surfaces.



Homework Equations


[tex]z^2=x^2+y^2[/tex] with plane [tex]z=1+y[/tex]


The Attempt at a Solution


So shouldn't it be
[tex]r(t)=<cos(t), sin(t), 1+sin(t)>[/tex]
since x=cos(t), y=sin(t), and z= 1+sin(t)?
The book gives a wacky answer
 
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  • #2
What kind of a shape do you think
[tex]z^2=x^2+y^2[/tex]
is?
 
  • #3
I know [tex]z^2=x^2+y^2[/tex] is a cone.
 
  • #4
Can anyone help me, I have got probably an hour
 
  • #5
:uhh::cry::cry::cry:
 
  • #6
The shape you parametrized is an elipse, but the intersection of the double-cone with side slope 1, and a plane with slope 1 is going to be a parabola.
 
  • #7
So how would I go about coming to a vector equation?
 
  • #8
Weave said:
So how would I go about coming to a vector equation?

Since you already have:
[tex]z=1+y[/tex]
I'd substitute, simplify, and see what happens:

[tex]z^2=x^2+y^2[/tex]
[tex](1+y)^2=x^2+y^2[/tex]
...

Which should work out reasonably well.
 
  • #9
Weave said:
...
So shouldn't it be
[tex]r(t)=<cos(t), sin(t), 1+sin(t)>[/tex]
since x=cos(t), y=sin(t), and z= 1+sin(t)?
The book gives a wacky answer

There are more than one way to parameterize a curve, so your answer and the books answer needn't agree.
[edit:]
But I note an error. Since you've chosen [itex]x=cos(t), y=sin(t)[/itex] then [itex] x^2+y^2=1[/itex] you've implicitly added then another constraint and you cannot satisfy [itex] z^2 = x^2+y^2[/itex]. Rather try [itex] x = z\cdot\cos(t)[/itex] and [itex]y=z\cdot\sin(t)[/itex]

[end edit:]
With regard to vectorizing you have already done that:
[tex]\mathbf{r}(t)=<x(t), y(t), z(t)> = x(t)\hat{\imath}+y(t)\hat{\jmath} + z(t)\hat{k}[/tex]
These are just two ways of writing the same vector. The basis is:
[tex]\langle 1,0,0\rangle=\hat{\imath}[/tex]
[tex]\langle 0,1,0\rangle = \hat{\jmath}[/tex]
[tex]\langle 0,0,1\rangle = \hat{k}[/tex]
 
Last edited:
  • #10
Weave said:

Homework Statement


Find a vector function that represents the curve of the intersection of two surfaces.



Homework Equations


[tex]z^2=x^2+y^2[/tex] with plane [tex]z=1+y[/tex]


The Attempt at a Solution


So shouldn't it be
[tex]r(t)=<cos(t), sin(t), 1+sin(t)>[/tex]
since x=cos(t), y=sin(t), and z= 1+sin(t)?
The book gives a wacky answer

Since z= 1+ y, z^2= (1+y)^2= 1+ 2y+ y^2= x^2+ y^2. Cancelling the y^2 on each side, 1+ 2y= x^2 or y= (x^2- 1)/2. Taking x itself to be the parameter, we have x= t (of course, y= (t^2- 1)/2, z= 1+ y= 1+ (t^2- 1)/2
 

1. What is parametrization and why is it important?

Parametrization is the process of representing a mathematical equation or function in terms of one or more variables, known as parameters. This allows for the manipulation and analysis of complex mathematical expressions, making it an essential tool in various fields of science, engineering, and mathematics.

2. How do I approach solving parametrization problems?

The first step in solving parametrization problems is to identify the parameters in the given equation. Then, you can set up a system of equations and use techniques such as substitution and elimination to find the values of the parameters. It is also helpful to visualize the problem graphically to gain a better understanding of the solution.

3. What are some common applications of parametrization?

Parametrization is used in a wide range of fields, including physics, biology, economics, and computer science. It is commonly used to model physical systems, such as the motion of objects and the behavior of fluids. In biology, it can be used to describe population growth and the spread of diseases. In economics, it is used to analyze supply and demand curves. In computer science, it is used in data visualization and computer graphics.

4. Can you demonstrate an example of parametrization in action?

Sure! Let's say we have the equation x = 2t^2 and y = t + 1, where t is the parameter. We can use these equations to plot a parabola in the xy-plane, where x is a function of t and y is a function of t. By varying the value of t, we can see how the parabola changes shape and position in the plane. This is just one example of how parametrization is used to visualize mathematical equations.

5. Are there any common mistakes to watch out for when working with parametrization?

One common mistake is forgetting to eliminate the parameter after solving the system of equations. This is important because the final solution should only be in terms of x and y, not t. Another mistake is not checking for any restrictions on the parameters, such as when working with trigonometric functions. It is also important to carefully define the domain and range of the parameter to ensure accurate solutions.

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