Answer
$ \frac{4}{5}i-\frac{3}{5}j, \frac{6}{5}i+\frac{8}{5}j$
Work Step by Step
1. Let the two required vectors be $\vec x=ai+bj, \vec y=ci+dj$, use the given conditions, we have $a+c=2, b+d=1,$
2. Let $\vec x\parallel \vec w$, with the same slope, we have $\frac{b}{a}=\frac{3}{-4}$, thus $3a=-4b,$
3. Let $\vec y\perp\vec w$, we have dot product $(-4)(c)+(3)(d)=0$, thus $4c=3d,$
4. Use the above results, we have $12a+12c=24$ or $-16b+9d=24$, plus $b+d=1$ or $d=1-b$,
5. Solve the above to get $-16b+9-9b=24$, thus $b=-\frac{3}{5}$ and $d=\frac{8}{5}$
6. Use the known relations to find the other values $a=\frac{4}{5}$ and $c=\frac{6}{5}$
7. Thus the two vectors are $\vec x=\frac{4}{5}i-\frac{3}{5}j, \vec y=\frac{6}{5}i+\frac{8}{5}j$
