Tangent vector on the intersection of surfaces

In summary, a tangent vector on the intersection of surfaces is a vector that represents the direction and rate of change of the surfaces at a specific point of intersection. It can be calculated by taking the partial derivatives of the surfaces at the point of intersection. The tangent vector is important because it helps us understand the behavior of the surfaces and can be used to determine the direction of the tangent plane and the normal vector. It can also change at different points of intersection due to varying rates of change and directions of curvature. The tangent vector is also parallel to the gradient vector of the surfaces at the point of intersection, but represents a specific rate of change while the gradient vector represents the overall direction of steepest increase.
  • #1
Mr Davis 97
1,462
44

Homework Statement


The surfaces ##x^2+y^2 = 2## and ##y=z## intersect in a curve ##C##. Find a unit tangent vector to the curve ##C## at the point ##(1,1,1)##.

Homework Equations

The Attempt at a Solution


So I'm thinking that we can parametrize the surfaces to get a vector for the curve ##C##.

Let ##z=t##. Then ##y=t##. Then ##x = \sqrt{2-t^2}##. So we have a vector for the curve ##C##, ##\vec{r} (t) = \langle \sqrt{2-t^2}, t,t \rangle##. Then ##\vec{r}' (t) = \langle \frac{-t}{\sqrt{2-t^2}}, 1,1 \rangle##, and ##\vec{r}' (1) = \langle -1, 1,1 \rangle##. Then re-scaling to get a unit vector, we get ##\langle \frac{-1}{\sqrt{3}}, \frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}} \rangle##. Is this the correct answer?
 
Physics news on Phys.org
  • #2
Mr Davis 97 said:

Homework Statement


The surfaces ##x^2+y^2 = 2## and ##y=z## intersect in a curve ##C##. Find a unit tangent vector to the curve ##C## at the point ##(1,1,1)##.

Homework Equations

The Attempt at a Solution


So I'm thinking that we can parametrize the surfaces to get a vector for the curve ##C##.

Let ##z=t##. Then ##y=t##. Then ##x = \sqrt{2-t^2}##. So we have a vector for the curve ##C##, ##\vec{r} (t) = \langle \sqrt{2-t^2}, t,t \rangle##. Then ##\vec{r}' (t) = \langle \frac{-t}{\sqrt{2-t^2}}, 1,1 \rangle##, and ##\vec{r}' (1) = \langle -1, 1,1 \rangle##. Then re-scaling to get a unit vector, we get ##\langle \frac{-1}{\sqrt{3}}, \frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}} \rangle##. Is this the correct answer?
I don't see anything wrong with your work.

It might be helpful to sketch a graph of the curve C. Since it is formed by the intersection of a right circular cylinder whose central axis is along the z-axis, with the plane y = z, the curve of intersection is an ellipse.
 
  • #3
Mark44 said:
I don't see anything wrong with your work.

It might be helpful to sketch a graph of the curve C. Since it is formed by the intersection of a right circular cylinder whose central axis is along the z-axis, with the plane y = z, the curve of intersection is an ellipse.
Also, one thing. What's the difference between letting ##x = -\sqrt{2-t^2}## and letting ##x = \sqrt{2-t^2}##?
 
  • #4
Mr Davis 97 said:
Also, one thing. What's the difference between letting ##x = -\sqrt{2-t^2}## and letting ##x = \sqrt{2-t^2}##?
You'll get one side or another of the ellipse I mentioned, which is symmetric across the y-z plane. With the positive root, you get points in the first octant (really, on the "front" side of the y-z plane; if z < 0, the curve isn't in the first octant any more). With the negative root, you get points on that back side of the y-z plane.
 

Related to Tangent vector on the intersection of surfaces

1. What is a tangent vector on the intersection of surfaces?

A tangent vector is a vector that is tangent to a surface at a specific point of intersection. It represents the direction and rate of change of the surface at that point.

2. How is a tangent vector calculated on the intersection of surfaces?

A tangent vector can be calculated by taking the partial derivatives of the surfaces at the point of intersection and using them as the components of the vector.

3. Why is the tangent vector important on the intersection of surfaces?

The tangent vector helps us understand the behavior of the surfaces at the point of intersection. It can also be used to determine the direction of the tangent plane and the normal vector at that point.

4. Can the tangent vector change at different points of intersection?

Yes, the tangent vector can change at different points of intersection as the surfaces themselves may have varying rates of change and different directions of curvature.

5. How is the tangent vector related to the gradient of the surfaces?

The tangent vector is parallel to the gradient vector of the surfaces at the point of intersection. This means that the tangent vector and the gradient vector have the same direction, but the tangent vector represents a specific rate of change while the gradient vector represents the overall direction of steepest increase.

Similar threads

  • Calculus and Beyond Homework Help
Replies
6
Views
620
  • Calculus and Beyond Homework Help
Replies
1
Views
277
  • Calculus and Beyond Homework Help
Replies
1
Views
574
  • Calculus and Beyond Homework Help
Replies
1
Views
540
  • Calculus and Beyond Homework Help
Replies
8
Views
581
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
670
  • Calculus and Beyond Homework Help
Replies
9
Views
883
  • Calculus and Beyond Homework Help
Replies
12
Views
1K
  • Calculus and Beyond Homework Help
Replies
10
Views
825
Back
Top