Showing the derivative of a vector is orthogonal to the vector

[davidp92]

Homework Statement

http://i.imgur.com/6j8W6.jpg
I'm trying to understand that example in the text. I can imagine a curve on a sphere having the derivative vector being orthogonal to the position vector. What I don't understand is, how does "if a curve lies on a sphere with center the origin" mean the same thing as "if |r(t)|=c (constant)"?

Homework Equations

The problem is I don't understand why the statement says "Show that if |r(t)|=c ..."
Doesn't the example mean that every derivative vector of a curve is orthogonal to the position vector since |r(t)|=c for all t as long as the curve is continuous? (thinking it in terms of sqrt(x^2+y^2+z^2))
When is |r(t)| not equal to a constant?

 

[lineintegral1]

If |r(t)| is a constant, then you are essentially talking about a radius. Think about what the definition of |r(t)| is (it looks like you have an idea in the relevant equations section). If it is not equal to a constant, then the position vector and the derivative are not necessarily orthogonal. It is easy to find an example of this (for example, r(t) = <t,t>).

 

[Dick]

I don't quite get your question. |r(t)|=c means the position r(t) is at a constant distance of c from the origin at all times. If it's always at a distance of c from the origin, then it lies on a sphere of radius c around the origin. For a general curve, the distance from the origin will vary with time.

 

[davidp92]

If |r(t)| is a constant, then you are essentially talking about a radius. Think about what the definition of |r(t)| is (it looks like you have an idea in the relevant equations section). If it is not equal to a constant, then the position vector and the derivative are not necessarily orthogonal. It is easy to find an example of this (for example, r(t) = <t,t>).

How is |r(t)| not equal to a constant for r(t)=<t,t>?

Thanks for replying!

 

[davidp92]

I don't quite get your question. |r(t)|=c means the position r(t) is at a constant distance of c from the origin at all times. If it's always at a distance of c from the origin, then it lies on a sphere of radius c around the origin. For a general curve, the distance from the origin will vary with time.

Ohh, I get it now. I wasn't thinking of it the way you have it in the bold.

Thanks!