Understanding the Unit Normal Vector in Multivariable Differential Calculus

In summary, the question asks about the proof for the equation __r'(t) x r''(t) x r'(t)_ = N(t) / ||r'(t) x r''(t) x r'(t)|| and the role of r and N in this equation. The answer suggests considering the geometric behavior of the cross product and the tangent and normal vectors in relation to the space curve defined by r(t).
  • #1
issisoccer10
35
0
This question comes from my multivariable differential calculus course. I cannot figure how to prove that the following is true...

How does...

__r'(t) x r''(t) x r'(t)_ = N(t) ?
||r'(t) x r''(t) x r'(t)||

any help would be appreciated...thanks
 
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  • #2
What are r and N?
 
  • #3
issisoccer10 said:
This question comes from my multivariable differential calculus course. I cannot figure how to prove that the following is true...

How does...

__r'(t) x r''(t) x r'(t)_ = N(t) ?
||r'(t) x r''(t) x r'(t)||

any help would be appreciated...thanks

Just think about how the cross product behaves geometrically. I'm assuming r is a map from R into R3 defining a space curve and that N is a normal vector to the curve, defined by being normal to the tangent vector at each point. r'(t) would then define a tangent vector at the point r(t). r''(t) describes the change in the tangent vector at that point, so for an arbitrarily small window, it should lie in the osculating plane of the curve. What direction does the cross product of these two vectors go in with respect to these two vectors? What then happens when you take the cross product of this vector with the tangent vector?
 

1. What is a unit normal vector?

A unit normal vector is a vector that is perpendicular to a curve or surface at a specific point, and has a magnitude of 1. It is used to describe the direction of the normal or perpendicular line to a curve or surface at a given point.

2. How is the unit normal vector calculated?

The unit normal vector is calculated by taking the derivative of the curve or surface at a specific point and then normalizing the resulting vector to have a magnitude of 1.

3. What is the significance of the unit normal vector in physics?

In physics, the unit normal vector is used to describe the direction and magnitude of force or acceleration acting on an object at a specific point on a curve or surface. It is also used in calculating surface integrals and flux in vector calculus.

4. Can the unit normal vector be negative?

Yes, the unit normal vector can have a negative direction, but its magnitude will always be 1. The direction of the unit normal vector depends on the orientation of the curve or surface at a specific point.

5. How is the unit normal vector used in computer graphics?

In computer graphics, the unit normal vector is used to determine the shading of a 3D object by calculating the angle between the unit normal vector and the light source. It is also used in creating smooth surfaces and determining the direction of reflections and refractions.

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