Understanding the Shape and Orientation of a Path in Vector Calculus

[calculusisrad]

Let c(t )=(2t,sint,cost) be a path. Describe the shape and orientation of this path

Describe the shape and orientation between points (0,0,1) and (pi,1,0)

I have no idea how to figure out the shape of a curve from its path and my book is only confusing me. Please help!

 

[tiny-tim]

hi calculusisrad!

Let c(t )=(2t,sint,cost) be a path. Describe the shape and orientation of this path

do the easy bit first …

ignore the x coordinate (ie look at it from the x direction, or project it onto a yz plane) …

what does it look like in 2D ? ​

 

[RoshanBBQ]

I would suggest, perhaps, exploring these functions in fewer dimensions than 3. For example, you are familiar with a plot of two coordinates like so:
x = t
y = f(t)
Plotting the above in an xy plot is the equivalent to y = f(x).

Maybe you're confused, because the graph of the answer does not have "output" versus "independent variable" like you usually have done. Instead, you have output on the x-axis, y-axis, and z-axis. The independent variable is not graphed.

Think to yourself along each axis separately and then fuse the results together in your head. I am assuming your vectors are <x, y, z>. What are the values of x doing as t marches upward? What are the values of y doing as t marches upward? What about the values of z? Can you at least plot an xy slice? Is the yz slice confusing you?

You're not really easy to help since you haven't stated your confusion.

Google complex exponential graph.

 

[calculusisrad]

Thanks you so much for responding! But you guys seems to just be telling me how to graph the function. But its a path, and paths map out the actual function in some weird way I don't understand. Don't I need to take that into account? I'm sorry, I just really don't understand paths at all. I could be completely wrong in my interpretation of the problem.

 

[tiny-tim]

hi calculusisrad!

… its a path, and paths map out the actual function in some weird way I don't understand. Don't I need to take that into account?

basically, a path is a curve: add an arrow to it to show the direction of increasing t

start with the 2D version in this case …

what does the path look like? ​