Position vector perpendicular to tangent vector yields a sphere

In summary, the problem requires finding a solution to the given equation where the position vector is perpendicular to the tangent vector, leading to the conclusion that the curve lies on a sphere with center at the origin. The solution to the problem involves using vector equations.
  • #1
physicsidiot1
6
0

Homework Statement


If a curve has the property that the position vector r(t) is always perpendicular to the tangent vector r'(t), show that the curve lies on a sphere with center the origin.


Homework Equations



-1/r'(t)= slope of position vector

x[tex]^{2}[/tex]+y[tex]^{2}[/tex]=1

The Attempt at a Solution


Not really sure where to begin for this one.
 
Physics news on Phys.org
  • #2
hi physicsidiot1! :smile:
physicsidiot1 said:
… the position vector r(t) is always perpendicular to the tangent vector r'(t) … the curve lies on a sphere with center the origin.

this is a vector problem, so it needs a vector solution :wink:

start by writing the question out in two vector equations …

what do you get? :smile:
 

1. What is a position vector and a tangent vector?

A position vector represents the location or position of a point in space, while a tangent vector represents the direction or slope of a curve at a certain point.

2. How can a position vector be perpendicular to a tangent vector?

A position vector can be perpendicular to a tangent vector if the curve is a sphere, as the tangent vector at any point on a sphere will always be perpendicular to the position vector from the center of the sphere to that point.

3. Why does a position vector perpendicular to a tangent vector yield a sphere?

This is because a sphere is a special type of curve where the tangent vector is always perpendicular to the position vector at any point on the sphere. Therefore, if the position vector is perpendicular to the tangent vector, it will always point towards the center of the sphere, creating a spherical shape.

4. What is the significance of a position vector perpendicular to a tangent vector in physics?

In physics, this concept is important for understanding motion and forces in circular or spherical systems. For example, the position vector and tangent vector can be used to calculate the acceleration of an object moving in a circular path.

5. Can a position vector and tangent vector be perpendicular in other shapes besides a sphere?

No, a position vector and tangent vector can only be perpendicular in a sphere. In other shapes, the tangent vector may not always be perpendicular to the position vector, as the direction and slope of the curve can vary at different points.

Similar threads

  • Calculus and Beyond Homework Help
Replies
8
Views
348
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
3K
  • Differential Geometry
Replies
21
Views
588
  • Calculus and Beyond Homework Help
Replies
1
Views
960
  • Calculus and Beyond Homework Help
Replies
5
Views
1K
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
14
Views
2K
Back
Top