So I show a case, when the row notation and the column notation should not be confused.
Lets take two vectors:
\(\vec{a}=\left(\begin{matrix}1\\2\end{matrix}\right)\) and \(\vec{b}=(3,4)\).
And now multiply \(\vec{a}\) on \(\vec{b}\), and then \(\vec{b}\) on \(\vec{a}\):
\(\vec{a}\vec{b}=\left(\begin{matrix}1\\2\end{matrix}\right)(3,4)=\left(\begin{matrix}3&4\\6&8\end{matrix}\right)\)
\(\vec{b}\vec{a}=(3,4)\left(\begin{matrix}1\\2\end{matrix}\right)=3\cdot1+4\cdot2=11\)
If we confuse the rows and the columns we will have the wrong result, because it is important to know where is the column and where is the row.
But in other case, when we say about a vector in general, without multiplication, it is not so important to know if it is a row or a column.