How do I know if two vectors with $n$ components are parallel?

Question

How do I know if two vectors with $n$ components are parallel?

For example

$$\begin{pmatrix}5\\2\\1\\3\\4\end{pmatrix} \text{, and } \begin{pmatrix}4\\1\\2\\3\\6\end{pmatrix}.$$

Answer 1

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.

Answer 2

Just check if $$ | v \bullet w| = |v| \cdot |w|, $$ where $\bullet$ denotes the scalar product. See here.

Answer 3

Denote the two vectors as $\alpha$ and $\beta$. Try the following steps:
(1) If one of the vectors is null, then they are parallel.
(2) If $\alpha$ has a component $0$, but the correspondent component of $\beta$ is not $0$, then the two vectors are NOT parallel.
(3) If $\alpha$ has a component $a_i \ne 0$, and the correspondent component of $\beta$ is $b_i$, take down the quotient $b_i/a_i$. If all the quotients you can get are the same, then the two vectors are parallel; otherwise, they are not parallel.

Answer 4

Put them together like: $$\begin{pmatrix} 5 & 4 \\ 2 & 1 \\ 1 & 2 \\ 3 & 3 \\ 4 & 6\end{pmatrix},$$ or in rows. If one of the $2 \times 2$ determinants there is non-zero, they're not parallel. They are not parallel because, for example, $\left|\begin{matrix} 5 & 4 \\ 2 & 1\end{matrix}\right| = -3 \neq 0$.