Understanding the Unit Normal Vector in Multivariable Differential Calculus

[issisoccer10]

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

 

[Office_Shredder]

What are r and N?

 

[slider142]

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?