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?
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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.
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.
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".
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