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

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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.

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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.

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