According to my textbooks, a unit vector has no units and no dimensions, but is only used to specify direction. It only shows the orientation of a corresponding vector in space. I think it's true, or that's what it looks like. It makes sense because a unit vector is defined as 'a vector' divided by its magnitude. Since we have the same numerical value in numerator and the denominator, a unit vector has a magnitude of 1 unit. Likewise, we have the same unit in both numerator and the denominator, that makes a unit vector 'unitless', and hence dimensionless. That's why I think a unit vector has no dimensions. Please correct me if I'm wrong.
But, another question naturally comes to our mind. Why if I say, "a force of 1 N due east" or "a displacement of 1m, 30° NOE"?
Both force and displacement are vector quantities, and both have a magnitude of 1 unit in the above two examples. My question is, can we call these two "unit vectors"? That's what I'm struggling to understand. There's no reason why we can't call these two unit vectors. Because both have a magnitude of 1 unit, and both are vectors. However, both have units, and hence both are not dimensionless.