Determine a point $$\text{ABC- triangle:} A(4,2); B(-2,1);C(3,-2)$$
Find a D point so this equality is true:
$$5\vec{AD}=2\vec{AB}-3\vec{AC}$$

So,let's observe picture below.first of all you will need to find point $E$...use that $E$ lies on $p(A,B)$ and that $\left\vert AB \right\vert = \left\vert BE \right\vert $. Since $ p(A,C)\left\vert  \right\vert p(F,E)$ we may write next equation: $\frac{y_C-y_A}{x_C-x_A}=\frac{y_E-y_F}{x_E-x_F}$ and $\left\vert EF \right\vert=3 \left\vert AC  \right\vert$ so we may find point F.Since $\left\vert AF \right\vert=5 \left\vert AD  \right\vert$ we may write next equations: $x_D=\frac{x_F+4x_A}{5}$ and $y_D=\frac{y_F+4y_A}{5}$

