# Why the direction of the omega (angular velocity vector) is along the axis of rotation? Also for angular acceleration

I know that the direction of omega is taken along the axis of rotation but I don't understand it why it is taken?

I also know that $$\mathbf v = \mathbf ω \times \mathbf r$$ so, all three vectors should be perpendicular, but this also doesn't satisfy me. As this is merely a formula but a formula don't give me that feel how that direction of omega(angular velocity vector) changes angular displacement of a body doing circular motion as this is also known that [Bold letters are vectors]

$$\boldsymbol{\omega} = \frac{\Delta\boldsymbol{\theta}}{\Delta t}$$

Just like velocity changes linear displacement of a body.

## 1 Answer

We can define the angle as the area sweep by a vector in the rotating plane with its strating point at the rotating position. As the vector rotates a angle $$\Delta \theta$$, the area swept by the vector is: $$\Delta A = r^2 \Delta \theta$$ Therefore, we may define the angle as the area swept divided by $$r^2$$. The area is a vector quantity with direction on the normal direction of the area. This defines the direction of the angle $$\omega$$. I will show that the definition of area is more general that the "angle" itself.

Case 1: Consider that the rotating vector is not ristricted in the plane, e.g. it rotates in a corrugated surface. The resultant rotation angle $$\theta$$ won't be the sum of the infinitismal angle $$\Delta \theta$$ $$\theta \neq \sum_i \Delta \theta_i$$ But it is equal to the sum of the infinitesmal area. $$\theta \text{ } \hat{\theta} = \sum_i \frac{\Delta \vec{A}_i}{r^2}$$ The $$\hat{\theta}$$ denotes the result vector direction of the vector summation.

Case 2: A solid angle $$\Omega$$ can only be defined using area $$\Omega = \int \frac{\hat{r} \cdot d\vec{A}}{r^2}$$

Hope this helps.