We develop the vector equation of the straight line $$r$$ expressed in coordinates: $$$\begin{array}{rcl}(x,y,z) &=& (a_1,a_2,a_3)+k\cdot (v_1,v_2,v_3) \\ (x,y,z) &=& (a_1,a_2,a_3)+ (k\cdot v_1,k\cdot v_2,k\cdot v_3)\\(x,y,z) &=& (a_1+k\cdot v_1,a_2+k\cdot v_2,a_3+ k\cdot v_3) \end{array}$$$ and by separating the coordinates we obtain: $$$\left.\begin{array}{rcl} x &=& a_1+k\cdot v_1 \\ y&=& a_2+k\cdot v_2 \\ z&=&a_3+k\cdot v_3\end{array}\right\}$$$ These are the parametric equations of the straight line.
Find the parametric equations of the straight line that goes through point $$A = (-1, 1, 3)$$ and that has $$\overrightarrow{v}=(3,-2,1)$$ as director vector.
The vector equation is $$$(x,y,z)=(-1,1,3)+k\cdot (3,-2,1)$$$ Separating by components we obtain: $$$\left.\begin{array}{rcl} x &=& -1+3k \\ y&=& 1-2k \\ z&=&3+k\end{array}\right\}$$$ which are the parametric equations.