Rotations as vectors
- Linear properties point backwards or forwards whereas
- angular properties “point” clockwise or counterclockwise.
While the backwards/forwards feeling is along something – a direction that defines it – what is the clockwise/counterclockwise feeling? Isn’t it rather around something?
Around what? Grab the screwdriver and turn it. Aren’t you actually rotating about the screwdriver itself? If you imagine a line along the screwdriver, then this is the line the angular properties act around. This line defines an angular motion, like a direction defines a linear motion.
Why don’t we name this imaginary line: an axis-of-rotation.
It made sense to think of position, velocity, acceleration and force as vectors $\vec s$, $\vec v$, $\vec a$, $\vec F$. Their vectors point the same way as the properties themselves act. If you know the vector, you know ‘how much’ and ‘which way’. But when something is rotating, how can we invent a vector to point “the same way”? Should that be a “curved” vector or so?
No, why don’t we simply decide that an angular-property vector points along the axis-of-rotation? Then, when writing $\vec \theta$, $\vec \omega$ and $\vec \alpha$,
- the magnitude still tells ‘how much’ but
- the direction of the vector tells ‘about which axis’.
Naturally, ‘angular acceleration is change in angular velocity’, which is ‘change in angular position’, also in their vector forms:
$$ \vec \alpha= \vec \omega’\qquad\qquad \vec \omega = \vec \theta’$$
Whenever you see an angular-property vector, remember that this vector is imaginary and has just been invented to indicate the axis. It does not represent the turning itself. It defines the turning but doesn’t represent it directly. Note that this vector clearly is perpendicular to the turning direction (an angular property acts around its axis and thus its vector).