How Do Curvature and Planes Relate in Vector Problems?

[nns91]

Homework Statement

1. Suppose r(t)= (e^t * cost) i + (e^t * sint) j. Show that the angle between r and a never change. What is the angle.

2. Find the equations for the osculating, normal, and rectifying planes of the curve r(t)=t i + t^2 j + t^3 k.

3. Express the curvature of a twice differentiable curve r= f(theta) in the polar terms of r and its derivatives

Homework Equations

Kappa, Torsion, cross product, dot product,...

The Attempt at a Solution

1. So a is the acceleration. Thus, it is the 2nd derivative. So do I find the normal vector of r and a and then take their cross product to find the angle ? What will be the normal vector to r ??

2. So is the osculating plane pretty much the curvature circle ? What is the rectifying plane ?

3. I am kinda lost in the problem. How should I attack this ?

 

[gabbagabbahey]

1. So a is the acceleration. Thus, it is the 2nd derivative. So do I find the normal vector of r and a and then take their cross product to find the angle ? What will be the normal vector to r ??

No, just take the second derivative. Do the unit vectors i and j ever change with time? If not, then the time-derivative of r is easy to find.

 

[nns91]

I took the 2nd derivative. But I need to find the angle though ?

 

[gabbagabbahey]

For any two vectors, u.v=||u||*||v||cos(theta)...where theta is the angle between them...so theta=___?

 

[nns91]

Right. I got it but how do I prove it that it does not change ??

 

[gabbagabbahey]

Well, what is d(theta)/dt?

 

[nns91]

0 ??

How about the 2nd problem ?

 

[nns91]

I think I got the first problem solved. How about the next 2 ?

 

[nns91]

For number 2, what is the equation of osculating and rectifying plane look like ? I know for the normal plane it is a(x-x0)+b(y-y0)+c(z-z0)=0

Is the rectifying plane formed by the binormal vector and the unit tangent vector T ??

Any suggestion for number 3 ?

 

[nns91]

I have already solved number 2.

Any suggestion for 3 ?