16. Given $\vec{a}=\{1,2\}, \vec{b}=\{-3,2\}$, find the real number $k$ such that $k \vec{a}+\vec{b}$ is in the same direction or opposite direction to $\vec{a}-3 \vec{b}$.
16. $k \vec{a}+\vec{b}=\{k-3,2 k+2\}, \vec{a}-3 \vec{b}=\{10,-4\}$. From the problem, we have $k \vec{a}+\vec{b} \parallel \vec{a}-3 \vec{b}$, so $\frac{k-3}{10}=$ $\frac{2 k+2}{-4}$, which gives $k=-\frac{1}{3}$.