12.14. (GDR, 74). Find all pairs of non-zero vectors $\boldsymbol{x}, \boldsymbol{y}$, for which the sequence of numbers

$$
a_{n}=|\boldsymbol{x}-n \boldsymbol{y}|(n \in \mathbf{N})
$$

is: a) increasing; b) decreasing.
12.14. Since $a_{n} \geqslant 0$ for all values of $n$, the increase or decrease of the sequence $\left\{a_{n}\right\}$ is equivalent to the increase or decrease of the sequence of numbers

$$
a_{n}^{2}=(x-n y)^{2}=x^{2}-2 n x y+n^{2} y^{2}
$$

which represents the sequence of values of a quadratic trinomial with a positive leading coefficient at points $n \in \mathbf{N}$. From the properties of quadratic trinomials, it follows that such a sequence cannot be decreasing, and it is increasing if and only if $a_{1}^{2} < 2 x y$, i.e., $3|y| > 2|x| \cos \varphi$

where $\varphi$ is the angle between the vectors $\boldsymbol{x}$ and $\boldsymbol{y}$, and condition b) is never satisfied.