Is a vector and a unit vector dimensionless

Question

Lets say I have a position vector $\vec r$. Is it dimensionless or does it have a dimension of length i.e $[L]$.

Also does the unit vector $\hat r$ have a dimension?

Answer 1

Vectors do have dimensions. Specifically, the dimension of a vector is (and always must be) the same as the dimension of its components. This also means that al the components of a vector must have the same dimension.

In your example, the position vector $\vec r$ does indeed have units of length.

The vector $\hat r$ is defined as $\hat r = \vec r / |\vec r|$. Since we know that both $\vec r$ and $|\vec r|$ have units of length, we can conclude that $\hat r$ has units $L / L = 1$. In other words, it is dimensionless.

Answer 2

Yes it does have the dimension $[L]$. You can, however, be more specific and assign a different length dimension to each component. So that the $x$ component has length dimension of $[L_x]$ and analogous for $y$ and $z$. These are called direction dimensions, and can be helpful in dimensional analysis.

Answer 3

I think it should have dimensions. Suppose the question were "what is the unit vector of 10 Newtons force pointing due north?". Then the answer is "1 Newton due north".