Isn't $\overrightarrow{\omega} t=\theta$ equating a vector to a scalar?

Question

Unless I'm misreading them somehow, my 1st year Mathematics BSc course notes use

$$\overrightarrow{\omega}\cdot t=\theta,\tag{1}$$

where

• $\overrightarrow{\omega}$ is angular velocity, a vector;
• $t$ is time, a scalar;
• $\theta$ is angle, a scalar.

But a vector times a scalar is a vector, so $\overrightarrow{\omega} \cdot t$ is a vector, so (1) equates a vector to a scalar. What's gone wrong?

Answer 1

Either you can make $\theta$ a vector, showing the axis of rotation as well as the amount, or you can take the magnitude of $\omega$ to make it a scalar equation. Which is appropriate depends on the rest of the argument.