Parametrical equations of the straight line

These are simply the vector equation separated by components: $(x, y) = (p_{1}, p_{2}) + k \cdot (v_{1}, v_{2})$ $\begin{array}{rcl} x & = & p_{1} + k \cdot v_{1} \\ y & = & p_{2} + k \cdot v_{2} \end{array}}$

Example

Find the parametrical equations of the straight line $r$ that crosses the points $(3, 4)$ and $(- 2, 6)$ .

The vector equation with $A = (3, 4)$ and $B = (- 2, 6)$ is: $(x, y) = A + k \cdot \vec{A B} = (3, 4) + k \cdot (- 5, 2)$ Therefore, the parametrical equations of the straight line are: $\begin{array}{rcl} x = 3 - 5 \cdot k \\ y = 4 + 2 \cdot k \end{array}}$