Question

write vector relation between angular velocity (w), tangential velocity (v), and position vector (r)

Answer 1

If you are asking the Vector Relation, then it will be as shown.

    Angular Velocity  Vector = tangential velocity vector × radius vector.

Mathematically, In symbols,

   ω (vector) = v (vector) × r (vector)

This will be correct relation.

Now there are some misconception s which i wants to discuss/ Like, we sometimes not apply vector sign on radius, but this will make our relation as wrong.

This is because, no vector sign means the magnitude and the magnitude of the radius is same, which means it is a constant.

This implies that the angular velocity vector is equal to the tangential velocity vector, which cannot be possible because direction of both are different, and vectors are only equal when they are equal in the magnitude as well as in the direction.

Thus it will be wrong.

Hope it helps.

Answer 2

Any quantity or an unit that has direction are called as vector units.

Here,

The angular velocity is directly proportional to the product of tangential velocity and position vector.

That is, the angular velocity is equal to the tangential velocity if and only if they are equal in magnitude and in direction.

The relationship,

Angular velocity = Tangential velocity * Position vector

w = v * r

The relationship holds true if and only if they are equal in magnitude and in direction.