how do I prove that a(t)*a'(t)=0; given that a(t) is a parametrized curve where the image of the curve lies on a sphere in R3?

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- Feb 15th 2011, 03:06 PM #1

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- Feb 15th 2011, 04:12 PM #2

- Feb 16th 2011, 06:13 AM #3

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I don't believe Prove It's proof is valid. He appears to be assuming that a is a real valued function when it should be vector valued. Also, he is trying to prove the

**converse**of the statement- that**if**the dot product is 0 then the curve lies in a sphere.

In fact, I don't believe the theorem, as stated, is true. Consider the curve . At any point on that curve, so the curve lies on the surface of the sphere with center at (0, 0, R) and radius R. But .

.

What**is**true is that if is a curve lieing on the surface of a sphere**with center at the origin**, then .

There are two ways to prove that

Analytically: Any curve lieing on the surface of a sphere with radius R and center at the origin can be written as

for some functions and

Then

Take the dot product of those two vectors.

Geometrically: , the position vector, for each t, is a vector from the origin, the center of the sphere, to the sphere. It is, of course, perpendicular to the tangent plane at that point. And , the tangent vector to a curve lieing in the surface of the sphere, is itself in the tangent plane.