Parametric equations of a straight line in the space

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: $\begin{array}{rcl} x & = & a_{1} + k \cdot v_{1} \\ y & = & a_{2} + k \cdot v_{2} \\ z & = & a_{3} + k \cdot v_{3} \end{array}}$ These are the parametric equations of the straight line.

Example

Find the parametric equations of the straight line that goes through point $A = (- 1, 1, 3)$ and that has $\vec{v} = (3, - 2, 1)$ as director vector.

The vector equation is $(x, y, z) = (- 1, 1, 3) + k \cdot (3, - 2, 1)$ Separating by components we obtain: $\begin{array}{rcl} x & = & - 1 + 3 k \\ y & = & 1 - 2 k \\ z & = & 3 + k \end{array}}$ which are the parametric equations.