rotation - Vector Nature Of Angular Velocity

I am currently reading about angular position, angular velocity, and angular acceleration. I came across this paragraph that was particularly confusing, and was wondering if someone could perhaps help me. Here is the paragraph:

"If a particle rotates in the $xy$ plane, the direction of $\vec{\omega}$ for the particle is out of the plane of the diagram when the rotation is counterclockwise and into the plane of the diagram when the rotation is clock-wise. To illustrate this convention, it is convenient to use the right-hand rule demonstrated in Figure 10.3. When the four fingers of the right hand are wrapped in the direction of rotation, the extended right thumb points in the direction of $\vec{\omega}$The direction of $\vec{\alpha}$ follows from its definition $\vec{\alpha}=d\vec{\omega}/dt$. It is in the same direction as $\vec{\omega}$ if the angular speed is increasing in time, and it is antiparallel to $\vec{\omega}$ if the angular speed is decreasing in time."

The above text in bold is what I am having most difficulty with. What do they mean? Also, how do I color words?

Answer

Say you're looking at the piece of paper on your desk; that is the $xy$ plane. You place a dot in the center of the paper; that's your origin.

Your angular momentum is $\vec{L}=\vec{r}{\times}m\vec{v}$. For this example, $\vec{\omega}$ points in the same direction as your angular momentum, because $\vec{L}=mr^2\vec{\omega}$.

The way I remember the directions for the right-hand rule are as follows:

• Thumb points up (this is $\vec{L}$, the cross product)


• Index finger points forward (this is $\vec{r}$, the left vector being crossed)


• Middle finger points to the left (this is $\vec{v}$, the right vector being crossed)

Let's say the particle is on the paper, directly above the origin, and moving counterclockwise. Then, $\vec{v}$ points to the left edge of the page. The direction of $\vec{r}$ goes from the origin to the point, so it points to the top of the page.

My thumb points up, so this is the direction of $\vec{L}$, and hence the direction of $\vec{\omega}$.

Crane your hand around so that your middle finger points right and your index finger continues to point to the top of the page, and your thumb points down, which is the direction of $\vec{\omega}$ for clockwise motion.