Find $c$ to show the following statement is $true$. $[c^2, c^3, c^4] \text { parallel to } [1,-2,4] \text { with the same direction }$

Question

$[c^2, c^3, c^4] \text { parallel to/same direction as } [1,-2,4]$

Find $c$ if it exists.

How do I see if they are parallel and find a c?

Generally, we can say that $r[1,-2,4] = [c^2, c^3, c^4]$

But then I have two unknowns and I don't know how to solve like this?

Answer 1

As $[c^2,c^2,c^4]=c^2[1,c,c^2]$ is parallel to and has the same direction as $[1,c,c^2]$, this amounts to solving the same problem for the latter. Actually, one can obviously find $c$ such that $[1,c,c^2]$ is not only parallel, but equal to $[1,-2,4]$.

Answer 2

Consider $$c=\frac {c^3}{c^2}=\frac {c^4}{c^3}=?$$