# Some vector relating problems

1. Mar 21, 2009

### nns91

1. The problem statement, all variables and given/known data

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

2. Relevant equations

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

3. 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 ?

2. Mar 21, 2009

### gabbagabbahey

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.

3. Mar 21, 2009

### nns91

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

4. Mar 21, 2009

### gabbagabbahey

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

5. Mar 21, 2009

### nns91

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

6. Mar 21, 2009

### gabbagabbahey

Well, what is d(theta)/dt?

7. Mar 21, 2009

### nns91

0 ??

How about the 2nd problem ?

8. Mar 21, 2009

### nns91

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

9. Mar 22, 2009

### 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 ?

10. Mar 22, 2009

### nns91

I have already solved number 2.

Any suggestion for 3 ?

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