Of course to know the direction of vector `omega` you allways need to know the direction of its defining vectors `v`and `R` , that is you need to know the direction of rotation (clockwise or not) and the plane of the rotation.

If you assume that the vector `omega` is always perpendicular to the plane of rotation (plane of vectors R and v),

`v = omega xx R` (1)

saying that the rotation is clockwise or anticlockwise is enough to determine also the direction of `omega` . This vector `omega` , always obeys the rule of the right screw: it has the direction of a right screw rotating the same way as the body rotates (clockwise, or anticlockwise). If the body rotates clockwise (to the right) the right screw enters the plane of rotation, thus omega is downwards into the plane of rotation. The figure is attached.

2) Actually this is the only way to determine the direction of the resultant vector from a cross product: it has the same direction as a right screw that rotates the first vector of the product to be parallel to the second vector of the product.

In the case of the relation (1) you just have the directions of the cross product `v ` and of one of the initial vectors `R` . This is enough to determine the direction of the angular velocity `omega` .

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