Because linear velocity is a vector which I can agree as it's because of the direction of the displacement but in case of angular velocity it's angles covered in certain time but angles got no direction then how come it got any direction

P.S I came to know it's something of a pseudo vector but didn't understand it so explain it in simple terms.Thanks


2 Answers 2


In the simplest case, a point mass or particle or any other suitable abstraction at ($\vec r$) moving (at velocity $\vec v$) through space (with an origin), one can define an angular velocity about the origin:

$$ \vec \omega = \vec r \times \vec v $$

where the cross product's three components are defined by:

$$ \omega_i = \epsilon_{ijk}r_jv_k $$

What's that? That's the same as defining and antisymmetric rank-2 tensor:

$$ \omega_{ij} = r_iv_j - r_jv_i ,$$

which has 3 independent components that transform under rotations just like an ordinary vector.

Under reflections (aka coordinate inversion, aka parity transformations), the angular velocity does not transform like a vector:

$$ \vec r \rightarrow -\vec r$$ $$ \vec v \rightarrow -\vec v$$

(that is, vectors are odd), while:

$$ \vec\omega \rightarrow +\vec\omega $$

The angular velocity "vector" is even, just like a rank-2 tensor. It is for this reason that it is called an axial vector.

Sometimes "axial-vector" is considered synonymous with "pseudo-vector", but their is a distinction: pseudo-vectors depend on the origin.

If I translate the orgin:

$$ \vec r \rightarrow \vec r + \vec a, $$

then $\vec \omega$ changes. Real vectors, like $\vec v$ and $\vec a$, don't do that. Of course, that leaves $\vec r$ out in the lurch, because it's not really a vector either, since it has an orgin which breaks translation symmetry. Really $\vec r$ is an affine point, and in any serious physics formula you're always talking about $\vec r - \vec r'$, which is a vector.


If a point moves in a circle , this circle lies on a plain, now if you want to describe tiis plain, you could give to vectors in this plain, you have many possibilities to chose them. or you take a single vector perpendicular to the plain and it describes the plain, so to dircribe the motion of the point in the plain, you give the absolute value of your angular velocity and say the direction is in the direction of the plain. you still have two direction, but one decides to take the direction a normal screw goes, when you turn it like your point turns. Tis is not so important for the vector of th angular velocity, bur for the angular momentum which stays in the same plain, see conservation of angular momentum .


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