How do I know if two vectors with $n$ components are parallel? How do I know if two vectors with $n$ components are parallel?
For example 
$$\begin{pmatrix}5\\2\\1\\3\\4\end{pmatrix} \text{, and } \begin{pmatrix}4\\1\\2\\3\\6\end{pmatrix}.$$

Check to see if they are constant multiples of one another. In your example, if there were a $c$ such that
$$c\begin{pmatrix}
5\\2\\1\\3\\4
\end{pmatrix}=
\begin{pmatrix}
4\\1\\2\\3\\6
\end{pmatrix}$$
then by equating the first components, we see that $c$ would have to be $\frac{4}{5}$. However, when we check the second coordinate, $\frac{4}{5}$ doesn't work, since $\frac{4}{5}\cdot2\neq1$. This proves that no such $c$ exists. Since they are not constant multiples of one another, they are not parallel.
