Space Curve Intersecting a Parabloid

  • Thread starter Thread starter k_squared
  • Start date Start date
  • Tags Tags
    Curve Space
Click For Summary
SUMMARY

The intersection of the space curve \(\vec{r}(t) = \langle t, 0, 2t - t^2 \rangle\) with the paraboloid defined by the equation \(z = x^2 + y^2\) can be determined by substituting the parametric equations into the paraboloid's equation. Specifically, substituting \(x = t\) and \(z = 2t - t^2\) leads to the equation \(2t - t^2 = t^2\). Solving this results in the intersection points at \(t = 0\) and \(t = 2\), which correspond to the points \((0, 0, 0)\) and \((2, 0, 0)\) on the paraboloid.

PREREQUISITES
  • Understanding of parametric equations
  • Knowledge of paraboloid equations
  • Ability to solve quadratic equations
  • Familiarity with vector notation in calculus
NEXT STEPS
  • Study the method of parameterizing surfaces in three-dimensional space
  • Learn how to find intersections of curves and surfaces
  • Explore solving quadratic equations in multiple variables
  • Investigate vector calculus applications in geometry
USEFUL FOR

Students in calculus or multivariable mathematics, particularly those studying vector functions and their intersections with surfaces.

k_squared
Messages
62
Reaction score
0

Homework Statement



At what point does the curve [itex]\vec{r}(t) = <t,0,2t-t^2>[/itex] intersects the paraboloid [itex]z=x^2+y^2[/itex]

Homework Equations



None Known

The Attempt at a Solution



I assume that it might be easier to parametrize [itex]z=x^2+y^2[/itex], but I'm not sure how to do that or if there's a more standard approach.

I missed a day of class and I can't find any examples more than vaguely similar to this in the book. :\


Thanks for any help!
 
Physics news on Phys.org
You can also try to de-parameterize ##\vec r(t)##. Hint: there won't be any y terms.
 

Similar threads

Quadratic Surfaces: Substitute (a,b,c) into z=y^2-x^2
  • · Replies 5 ·
Replies
5
Views
2K
Parametrize the Curve of Intersection
  • · Replies 4 ·
Replies
4
Views
3K
Tangent vector on the intersection of surfaces
  • · Replies 3 ·
Replies
3
Views
2K
Find all points of intersection
  • · Replies 6 ·
Replies
6
Views
2K
Reparameterizing a curve in path length parameter
  • · Replies 1 ·
Replies
1
Views
2K
How to identify the curve at intersection of level surfaces
  • · Replies 22 ·
Replies
22
Views
2K
Finding the parametric equation of a curve
  • · Replies 2 ·
Replies
2
Views
2K
Finding integral curves of a vector field
  • · Replies 3 ·
Replies
3
Views
2K
What is the shape of the curve z = x^2 + 2y^2 and how can it be sketched?
  • · Replies 4 ·
Replies
4
Views
2K
Parametrization of a curve(the intersection of two surfaces)
  • · Replies 7 ·
Replies
7
Views
5K