Answer
$\vec v_1=-\frac{1}{5}i-\frac{2}{5}j, \vec v_2=\frac{6}{5}i-\frac{3}{5}j$
Work Step by Step
1. Let $\vec v_1=ai+bj, \vec v_2=ci+dj$, we have $a+c=1, b+d=-1$
2. With $v_1\parallel w$, we have $\frac{b}{a}=\frac{-2}{-1}$, thus $2a=b$
3. With $v_2\perp w$, we have $c(-1)+d(-2)=0$, thus $c=-2d$
4. Combine the above results, we have $b-4d=2$ and $b+d=-1$, thus $b=-\frac{2}{5}$ and $d=-\frac{3}{5}$
5. Thus the two vectors are $\vec v_1=-\frac{1}{5}i-\frac{2}{5}j, \vec v_2=\frac{6}{5}i-\frac{3}{5}j$
