\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
Consider the following system involving products of variables:$$\begin{cases}\dfrac{dx}{dt} = xy + 1 \\ \dfrac{dy}{dt} = y^2 - 3x\end{cases}$$
  1. Find the velocity vector \(\begin{pmatrix} \dot{x} \\ \dot{y} \end{pmatrix}\) at the point \((1, 2)\).
    \( \begin{pmatrix}\input{143966}{}{3em}{2em}{}{} \\ \input{243966}{}{3em}{2em}{}{}\end{pmatrix} \)
  2. Find the velocity vector at the point \((2, -1)\).
    \( \begin{pmatrix}\input{343966}{}{3em}{2em}{}{} \\ \input{443966}{}{3em}{2em}{}{}\end{pmatrix} \)