Find if a Vecotr field is perpendicular to a curve

In summary, for Question #4, to determine if a vector field is perpendicular to a curve, you can take the dot product of the vector with the tangent vector to the curve at a given point. To generalize this to a vector field, you must check that the scalar product of the tangent vector of the curve and the vector field vanishes at every point of the curve. If the curve is only given implicitly, you may need to use the implicit function theorem to find the tangent vector. Another method to find the tangent direction is to take the cross product of the normals of the surfaces defined by the two equations.
  • #1
swraman
167
0

Homework Statement



http://math.berkeley.edu/~teleman/53f08/review2.pdf
Question #4

Homework Equations



Not sure

The Attempt at a Solution



I have no idea...
I really don't care about the answer, I have that, but I just don't know how to find out if a vector field is perpendicular to a curve.
Thanks
 
Last edited by a moderator:
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  • #2
Do you mean perpendicular at every point on the curve? You determine whether a single vector is perpendicular to a curve at a given point by taking the dot product of the vector with the tangent vector to the curve at that point. Can you generalize that to a vector field?
 
  • #3
If you want to find out if a vector field is perpendicular you have to check that the scalar product of the tangent vector of the curve and the Vector Field vanishes at every point of the curve.

Since you have given the curve only implicitly you would probably need implicit function theorem to actually get the tangent vector.
 
  • #4
The easy way to get a tangent direction is to take the cross product of the normals of the surfaces defined by the two equations.
 

Related to Find if a Vecotr field is perpendicular to a curve

1. What is a vector field?

A vector field is a mathematical concept that assigns a vector (which has both magnitude and direction) to every point in a given space.

2. How do you determine if a vector field is perpendicular to a curve?

To determine if a vector field is perpendicular to a curve, you can use the dot product. If the dot product between the vector field and the tangent vector of the curve is equal to zero, then the vector field is perpendicular to the curve.

3. What is the significance of a vector field being perpendicular to a curve?

A vector field being perpendicular to a curve means that the vector field is tangent to the curve at every point. This can be useful in applications such as fluid dynamics and electromagnetism.

4. Can a vector field be perpendicular to multiple curves?

Yes, a vector field can be perpendicular to multiple curves. This means that the vector field is tangent to all of the curves at every point where they intersect.

5. How is the concept of perpendicularity between a vector field and a curve used in real life?

The concept of perpendicularity between a vector field and a curve is used in various fields such as engineering, physics, and computer graphics. For example, it can be used to calculate the force of a fluid on an object, or to create realistic 3D models of objects and their movements.

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