Answer
$ \frac{27}{10}i+\frac{9}{10}j, -\frac{7}{10}i+\frac{21}{10}j$
Work Step by Step
1. Let the two required vectors be $\vec x=ai+bj, \vec y=ci+dj$, we have $a+c=2, b+d=3,$
2. Let $\vec x\parallel \vec w$, we have $\frac{b}{a}=\frac{1}{3}$, thus $a=3b,$
3. Let $\vec y\perp\vec w$, we have dot product $(3)(c)+(1)(d)=0$, thus $d=-3c,$
4. Use the above results, we have $3a+3c=6$ or $9b-d=6$, plus $b+d=3$,
5. Solve the above to get $10b=9$, thus $b=\frac{9}{10}$ and $d=\frac{21}{10}$
6. Find other values $a=\frac{27}{10}$ and $c=-\frac{7}{10}$
7. Thus the two vectors $\vec x=\frac{27}{10}i+\frac{9}{10}j, \vec y=-\frac{7}{10}i+\frac{21}{10}j$
