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When a body (in pure rotation) rotates along an axis passing through it, why do we represent the axis of rotation in vectorial notation? Wouldn't it be sensible enough to represent the angular velocity (or similar quantities) in the vectorial notation? Is it because the direction changes continuously and thus representing the vectors themselves, would be anomalous because a different vector in a way would be called as a vector similar to another?

AccidentalFourierTransform
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Tamoghna Haldar
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  • I don't understand what you are getting at. The axis of rotation has a direction so it's natural to represent it as a vector. This vector coincides with the direction of the angular velocity, but why should that make any difference? – John Rennie May 19 '16 at 17:03

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It is vector because it has a magnitude and a direction. Also typically the components of $\vec{\omega}$ are evaluated based on an inetial coordinate frame and thus only represent the motion of the body and not of the measuring frame.

  • Rotational speed $\omega = \| \vec{\omega} \|$
  • Direction of rotation $\hat{z} = \frac{\vec{\omega}}{\omega}$

Additionally, if the linear velocity of the rigid body $\vec{v}$ is given at some point, the location of the instant center of rotation is given by $$\vec{r}_{ICR} = \frac{\vec{\omega} \times \vec{v}}{\| \vec{\omega} \|^2} $$

So there is a lot of information given about the motion of the body from the 3 components of $\vec{\omega}$.

John Alexiou
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