Vector Valued Function and values of t parallel to the xy-plane

In summary: Using the given equation for r'(t), we can set the z component equal to zero and solve for t. This will give us all the values of t for which r'(t) is parallel to the x-y plane.
  • #1
Starlit_day
10
0

Homework Statement


So, the problem is this:
Find all values of t such that r'(t) is parallel to the xy-plane.
And my equation is:
r(t)=(Squareroot(t+1) , cos(t), t4-8t2)


Homework Equations



Well, I will definitely have to know how to take the dirivative of the given vector valued function, which I know how to do. I think An important thing to know would be: the cross-product of the two vector-valued functions. But I can't really think of any pertinant formulae off hand that would help with this problem ... (maybe that's why I'm confused).

The Attempt at a Solution


Okay, so as I said I need to take the derivative and then I thought take the cross-product of the derivative and the original function.
So I did that and got: r'(t)= (1/2(t+1)-1/2, -sin(t), 4t3- 16t)
Wasn't much of a problem there, but now I'm kinda clueless as to how I would find ALL values of t for r'(t) parallel to the xy-plane. So I was thinking about the cross-product because that will tell me if the vector valued function and its derivative are parallel (if I get 0 as my answer when I do the cross-product) then maybe I can figure out a parametric equation for a plane that contains the vector function (the first one not the derivative), and all values in that plane would be parallel to the derivative because the plane is parallel to the derivative. >.< That sounds kind of complicated and somewhat off base to me. I'm speculating here because I really am not sure what to do. Although, that really wouldn't help me much considering I have to deal with the xy-plane. Hm,
So does anyone think they could help steer me in the right direction conceptually? I think I'm stuck conceptually on how I would go about finding these values of t ... I mean I know what they mean by a Vector Valued Function I understand what a function is and what a vector is I know about derivatives and understand all of that and I know when two vectors are parallel when given the corrodinates, but I just don't get this problem. Could someone help me at least start thinking in the right direction.
2
 
Physics news on Phys.org
  • #2
Starlit_day said:

Homework Statement


So, the problem is this:
Find all values of t such that r'(t) is parallel to the xy-plane.
And my equation is:
r(t)=(Squareroot(t+1) , cos(t), t4-8t2)


Homework Equations



Well, I will definitely have to know how to take the dirivative of the given vector valued function, which I know how to do. I think An important thing to know would be: the cross-product of the two vector-valued functions. But I can't really think of any pertinant formulae off hand that would help with this problem ... (maybe that's why I'm confused).

The Attempt at a Solution


Okay, so as I said I need to take the derivative and then I thought take the cross-product of the derivative and the original function.
So I did that and got: r'(t)= (1/2(t+1)-1/2, -sin(t), 4t3- 16t)
Wasn't much of a problem there, but now I'm kinda clueless as to how I would find ALL values of t for r'(t) parallel to the xy-plane. So I was thinking about the cross-product because that will tell me if the vector valued function and its derivative are parallel (if I get 0 as my answer when I do the cross-product) then maybe I can figure out a parametric equation for a plane that contains the vector function (the first one not the derivative), and all values in that plane would be parallel to the derivative because the plane is parallel to the derivative. >.< That sounds kind of complicated and somewhat off base to me. I'm speculating here because I really am not sure what to do. Although, that really wouldn't help me much considering I have to deal with the xy-plane. Hm,
So does anyone think they could help steer me in the right direction conceptually? I think I'm stuck conceptually on how I would go about finding these values of t ... I mean I know what they mean by a Vector Valued Function I understand what a function is and what a vector is I know about derivatives and understand all of that and I know when two vectors are parallel when given the corrodinates, but I just don't get this problem. Could someone help me at least start thinking in the right direction.
2

For r'(t) to be parallel to the x-y plane, its z component has to be zero.
 

FAQ: Vector Valued Function and values of t parallel to the xy-plane

1. What is a vector valued function?

A vector valued function is a mathematical function that assigns a vector, rather than a scalar, to each value in its domain. It is typically denoted by the variable f and written as f(t) = <x(t), y(t), z(t)>, where x(t), y(t), and z(t) are scalar functions of t.

2. What is the significance of a vector valued function?

A vector valued function allows us to describe the path or trajectory of an object in three-dimensional space. It is commonly used in physics, engineering, and other fields to model the motion of particles, fluids, and other systems.

3. What does it mean for a vector valued function to be parallel to the xy-plane?

A vector valued function is parallel to the xy-plane if its z-component is always equal to zero. In other words, the function only varies in the x- and y-directions, and does not change in the z-direction.

4. How can I determine the values of t for which a vector valued function is parallel to the xy-plane?

To determine the values of t for which a vector valued function is parallel to the xy-plane, you can set the z-component of the function equal to zero and solve for t. This will give you the specific t-values at which the function is parallel to the xy-plane.

5. Are there any real-world applications of vector valued functions parallel to the xy-plane?

Yes, there are many real-world applications of vector valued functions parallel to the xy-plane. For example, in physics, a projectile launched at a specific angle will follow a path that is parallel to the xy-plane. In engineering, the motion of a car or airplane can also be described using vector valued functions parallel to the xy-plane. Additionally, in computer graphics and animation, vector valued functions are used to create 3D images and animations.

Back
Top