Angular Velocity is Vector Quantity
Theorem
The physical quantity that is angular velocity can be correctly handled as a vector.
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Proof
In order to show that angular velocity is a vector, it is sufficient to demonstrate that it fulfils the vector space axioms.
Specifically, all we need to do is demonstrate the following.
Let $\bsomega_1$ be the angular velocity of a body about an axis which passes through a fixed point $O$.
Let $P$ be a point in $B$ whose position vector with respect to $O$ is $\mathbf r$.
The tangential velocity of $P$ is $\bsomega_1 \times \mathbf r$.
Now let $\bsomega_2$ be the angular velocity about a different axis which also passes through $O$ at an angle to the first axis.
The tangential velocity of $P$ due to this angular velocity is $\bsomega_2 \times \mathbf r$.
But linear velocities combine as vectors.
Hence the total linear velocity of $P$ is given by:
| \(\ds \bsomega_1 \times \mathbf r + \bsomega_2 \times \mathbf r\) | \(=\) | \(\ds \paren {\bsomega_1 + \bsomega_2} \times \mathbf r\) | ||||||||||||
| \(\ds \) | \(\equiv\) | \(\ds \bsomega \times \mathbf r\) |
Hence the motion of $P$ is the same as it would be due to an angular velocity which is the vector sum of the two components.
$\blacksquare$
Sources
- 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text {IV}$: The Operator $\nabla$ and its Uses: $5$. Simple Examples of Curl (footnote $*$)