Quote by joeybenn
The problem I see is that the vector for say [tex]\omega[/tex] in a spinning bicycle wheel, would be constantly changing in the [tex]\hat{i}[/tex] and [tex]\hat{j}[/tex] components.

The direction of the vector [tex]\omega[/tex] specifies the axis of rotation and the direction of rotation by the righthand rule. If the speed of rotation isn't changing, and the orientation of the axis isn't changing, [tex]\omega[/tex] isn't changing.
Angular displacement is not a vector, despite having magnitude and direction, as it does not obey the commutative law for vectors: if you rotate the earth 90 degrees north and then 90 degrees east, it is not the same thing as rotating 90 degrees east and then 90 degrees north. However, for small angular displacements [tex]d\vec{\theta}[/tex] it obeys the commutative law approximately, and can be considered a vector: if you rotate the earth such that you move north 10 miles and then east 10 miles, this *is* about the same as moving east 10 miles and then north 10 miles. Thus, it's time derivative [tex]\vec{\omega} = d\vec{\theta}/dt[/tex] is a vector, and so is [tex]\vec{\alpha}[/tex].