Question

Prove of vector relation between velocity and angular velocity

Answer 1

Answer:

Explanation:

Well, both angular velocity and vectors are human constructs, so what you want is not really a proof, but a way to justify representing angular velocity with a vector.

We use vectors to represent quantities that have both a magnitude (such as mass) and direction (such as in physical space). Describing linear velocity as a vector has a highly intuitive quality. We perceive an object as moving either fast or slow, and in some direction. However, the magnitude can be something esoteric such as dollar value, and the direction could be long 5 axes of production costs.

So what you want is a way to justify that angular velocity has

A particular orientation in space which we will call the “direction”

A quantity associated with it that is independent of orientation

If you delve into the mathematics of rotation, you will find that when you rotate an object just a tiny bit (we say infinitesimally), then you can mathemtically describe that rotation in terms of a unique plane in which the rotation takes place, and an angle representing the amount of rotation.

Alternatively, we can describe the rotation in terms of a unique vector. The magnitude of the vector is the amount of rotation, and the direction is perpendicular to the plane of rotation. However, we could take the direction to be pointing to one side of the plane or the other. By convention we choose the direction using the right-hand-rule (RHR). Curl your fingers in the direction of rotation, and your thumb points the direction of the rotation vector.

Because of the way we represent space when we create mathematical models in physics, it turns out to be very convenient to give angular quantities (velocity, acceleration, torque, etc.) a vector representation as these vectors follow nice rules in terms of composition and physical interactions.

In physics we also use many other representations of rotation including matrices, tensors, quaternions, and others.

While you were specifically referring to the “Axis-Angle” representation, all of these representations are correct and useful.