Mathematica does not have the concept or row or column vectors like you may be used to. The concept isn't really necessary either and is just a convention to visualize the dot product (although I know there are people that vehemently object to this statement). 

In dot products like $M\cdot\vec{x}$ and $\vec{x}\cdot M$ Mathematica uses $\vec{x}$ as your column and row vector automatically. Both are entered in Mathematica as the same list.

    {{1, 2}, {3, 4}}.{5, 6}
    (* {17, 39} *)
    
    {5, 6}.{{1, 2}, {3, 4}}
    (* {23, 34} *)

Of course, this means that  the length of the vector must be equal to the number of columns in the matrix in the former case, and to the number of rows in the latter case.

So, this doesn't work (2x3 matrix):

    {{1, 2, 1}, {3, 4, 1}}.{5, 6}
    
  >  During evaluation of Dot::dotsh: Tensors {{1,2,1},{3,4,1}} and {5,6} have incompatible shapes. >>

    (* {{1, 2, 1}, {3, 4, 1}}.{5, 6} *)

and this works (3x2 matrix):
   
    {{1, 2}, {3, 4}, {1, 1}}.{5, 6}
    (* {17, 39, 11} *)

If you insist on using column and row vectors, you can *mimic* them by using similar looking *matrices*. The matrix equivalent of a column vector would be `{{x1}, {x2},...,{xn}}`. The equivalent of a row vector would be `{{x1, x2,...,xn}}`. If you use `MatrixForm` or `TableForm` they display just like column and row vectors would, except that they are still matrices in disguise (nx1 and 1xn ones, respectively). Those matrices behave just like their row and column vector brethren would in languages that have them, but remember, *they are not necessary at all in Mathematica*.

    {{1, 2}, {3, 4}}.{{5}, {6}}
    (* {{17}, {39}} *)
    
    {{5, 6}}.{{1, 2}, {3, 4}}
    (* {{23, 34}} *)

The above code typeset in `TraditionalForm` (just <kbd>Ctrl</kbd>+<kbd>Alt</kbd>+<kbd>T</kbd> of the above lines) displays as:

![Mathematica graphics](https://i.sstatic.net/LXUqU.png)