If $a\vec{x}+b\vec{y}=c\vec{x}+d\vec{y}$ then $a=c$ and $b=d$
Solution 1:
Advice from the author: Get in the habit of checking to see if you can reduce part b. of a problem to part a. This is how mathematics frequently works. In part a. you showed that if $sx+ty=0$ and $x$ and $y$ are non-parallel, then you must have $s=t=0$. Do not try to reinvent the wheel every time. What elementary algebra can you do to make this form appear? (You already did it to get to what you have.)