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## The direction of the angular velocity vector

In this blog, I introduced the idea of angular velocity, which is the rotational equivalent of linear velocity. The angular velocity $\omega$ is usually measured in radians per second, where radians are the more natural measurement of an angle than the more familiar degrees. But, just as linear velocity is a vector and therefore has a direction, so too does angular velocity. So, what is the direction of the angular velocity vector?

## The relationship between linear velocity and angular velocity

As we saw in the blog where I introduced the concept of angular velocity, it can be defined as simply the angle $\theta$ moved per unit time, or

$\omega = \frac{ \theta }{ t }$

which of course leads to it being usually measured in radians per second. However, we can also write the angular velocity in terms of the linear velocity. To see how to do this let us remind ourselves of the definition of a radian, the more natural unit for measuring an angle. As I introduced it in this blog, measuring an angle in radians just means dividing the length of the arc $l$ by the radius of the circle $r$.

An angle measured in radians is simply the length of the arc $l$ divided by the radius $r$

We can write that the angle measured in radians is

$\theta \text{ (in radians) } = \frac{ l }{ r }$

But, the linear velocity $v$ is just defined as distance divided by time, so we can write

$v = \frac{ l }{ t }$

Re-writing $l$ in terms of $\theta \text{ and } t$ we can write

$v = \frac{ \theta r }{ t }$

and so we can write the angle $\theta$ as

$\theta = \frac{ v t }{ r }$

Using this for $\theta$ we can write the angular velocity $\omega$ as

$\omega = \frac{ v t }{ r } \cdot \frac{ 1 }{ t }, \text{ so } \boxed{ \omega = \frac{ v }{ r } }$

## The direction of the angular velocity vector $\vec{ \omega }$

Writing this in terms of vectors, and remembering that division of vectors is not defined, we instead write that

$\boxed{ \vec{ \omega } = \frac{ \vec{ r} \times \vec{ v } }{ | \vec{ r } |^{2} } }$

where $\vec{ r } \times \vec{ v }$ is the vector product (or cross-product), as I discussed in this blog here.

The direction of the radius vector $\vec{ r }$ is away from the centre of the circle, and the direction of the linear velocity $\vec{ v }$ for an object moving anti-clockwise is in the direction shown in the diagram below, tangential to the circle so at right angles to the radial vector $\vec{ r }$.

The direction of the radius vector $\vec{ r }$ is away from the centre of the circle, the direction of the linear velocity $\vec{ v }$ for an object moving anti-clockwise is as shown, at right angles to the radius vector.

To find the direction of $\vec{ \omega }$, we can use the right-hand rule, as shown in the figure below.

The right-hand rule for determining the direction of the result of the vector product

In our example here, our first-finger is in the direction of the radial vector $\vec{ r }$, and our second-finger is in the direction of the linear velocity $\vec{ v }$, leading to the angular velocity $\vec{ \omega }$ (represented by the thumb) being outwards, or towards us, as shown in the figure below.

The direction of the angular velocity vector $\vec{ \omega}$ is perpendicular to the plane of rotation of the object.

Another way to find this direction is to wrap the fingers of the right hand in the direction of the rotation, the thumb will then show the direction of the angular velocity vector.

The direction of the angular velocity can also be found as shown in this figure.