<s>[INST] how to determine the conditions that 2 vectors parallel? Given 2 vectors,$u=(3,5)$,$v=(s,s^2)$,in what situations do u and v parallel?$(s≠0)$
In order to be parallel,$u$ must be proportional to $v$,vice verse.Let $k$ be a scalar $neq 0$,then $ku=(3k,5k)=v=(s,s^2)$,which gives $3k=s$;$5k=s^2 \rightarrow 5k=9k^2 (k \neq 0) \rightarrow 5=9k→k=\frac{5}{9}$,plug $k=\frac{5}{9}$ in the above equation $3k=s,s=\frac{5}{3}$.
Actually $k$ can also be multiplied to $s$,which is $kv=(ks,ks^2)=(3,5)$.Following the similar procedure,we get :$ks=3$;$ks^2=5 \rightarrow 3s=5,s=\frac{5}{3}$.But if we plug $s=\frac{5}{3}$ in $ks=3$,we get $\frac{5}{3k}=3$,which gives $k=\frac{9}{5}$.
The two methods should give the same answer.  I just don't know what's going wrong here.
 [/INST] There's nothing wrong. Your first value of $k = \frac{5}{9}$ means that $\frac{5}{9} u = v$. Your second value of $k = \frac{9}{5}$ is for k multiplying the other vector, and means that $\frac{9}{5} v = u$. But that's exactly the same you got before.
$$\frac{5}{9} u = v \implies u = \frac{9}{5} v$$
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