Vector functions: solving for curves of intersection

In summary, using a parametric representation allows you to obtain the same answer as that of the book using x = t.
  • #1
kylera
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Homework Statement


Solve for the vector function that represents the curve of intersection in the following two surface: z = sqrt(x^2 + y^2) and z = 1 + y


Homework Equations





The Attempt at a Solution


Through blind trial and error, I managed to get the book-specified answer of x = t, y = 0.5(t^2-1) and z = 0.5(t^2+1). What I'm curious about is why using parametric equations wouldn't work in this case. Which leads to what kind of right I have to using seemingly random functions when parameterizing a vector function.

I first worked with x = cos(t) and y = sin(t) to get z = 1, only to get something like 1 = 1 + sin(t), which got me stuck. Only after making some wild assumption x = t did I get the exact same book answer.
 
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  • #2
I don't understand what you mean by "using parametric equations". To solve this, all you need do is to equate the variables of both equations of surface. In this case, since they are both written as z = (something), simply equate the two of them. The resulting is the equation of curve you need.

Now this is where the parametric form comes in. We let x be the parameter, and we express y in terms of the parameter. This is why we assume x = t, and obtain the same answer as that of the book. Now of course no one says that we can't use some fancy parametric representation like x = 1-t or some other x = f(t) so long as f(t) isn't bounded above or below by real number, the way sin t is. The key thing is that the parameter must be allowed to vary across all possible real numbers for some values of t.
 
  • #3
Defennder said:
Now of course no one says that we can't use some fancy parametric representation like x = 1-t or some other x = f(t) so long as f(t) isn't bounded above or below by real number, the way sin t is. The key thing is that the parameter must be allowed to vary across all possible real numbers for some values of t.

In other words, when I try to obtain a function, it has to be able to return any real number value for some value t? Hence, x = t would work?
 
  • #4
x = t works because we usually assume [tex]t \in R[/tex] without having to write that our explictly (though you should). The "parameter function" f(t) has to be able to take on the all possible values of x.

For example, consider the unit circle at the origin. x^2+ y^2 = 1. A parametric representation of this is:
x = cos t
y = sin t.

We know that this works because cos t can take on any value betwee 1 and -1, just like sin t. But more importantly we know that his parametric representation satisfies the equation of the circle. Of course no one says you can't use a parametric representation like:

[tex]x = \sin (2t)[/tex]
[tex]y = \cos (2t)[/tex], just that in this case t isn't the angle between the x-axis and the line from the point (x,y) and the origin. Both these parametric representations satisfy the equation of the circle as well.
 

1. What are vector functions?

Vector functions are mathematical expressions that describe a vector in terms of a variable. They can be used to represent curves, surfaces, or other geometric objects in three-dimensional space.

2. How are vector functions used to solve for curves of intersection?

To solve for curves of intersection, we can set up a system of equations using the vector functions of the two curves. By finding the values of the variable that satisfy both equations, we can determine the points where the two curves intersect.

3. What is the importance of solving for curves of intersection?

Solving for curves of intersection is important in many areas of science and engineering, as it allows us to find the common points between two objects or systems. This can help us understand their relationship and make predictions about their behavior.

4. What are some techniques for solving for curves of intersection?

Some common techniques for solving for curves of intersection include substitution, elimination, and graphing. These methods involve manipulating the equations to eliminate one variable and solve for the remaining variable.

5. How can vector functions be visualized to understand curves of intersection?

Vector functions can be graphed in three-dimensional space to visualize how they intersect with each other. This can help us gain a better understanding of the relationship between the two curves and how they intersect at different points.

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