The parametric equations for a line are derived from their vector equation. Suppose you have a line in 3-space with vector equation r = r0 + td. Write all the vectors involved in terms of their components:

r = xi + yj + zk

r0 = x0i + y0j + z0k

d = ai + bj + ck.

Then the equation r = r0 + td becomes

xi + yj + zk = (x0i + y0j + z0k) + t(ai + bj + ck)

i.e. (rearrange and collect components together)

xi + yj + zk = (x0 + at)i + (y0 + bt)j + (z0 + ct)k.

Since equal vectors must have equal components, we have

x = x0 + at
y = y0 + bt
z = z0 + ct.

These equations are called the parametric equations for the line.

Parametric equations for a line give the coordinates of a generic point (x, y, z) on the line in terms of the coordinates of an initial point (x0, y0, z0) and a direction vector [a, b, c] . You get different points on the line for different values of the parameter t.

Notice that the numbers multiplying the parameter t in the parametric equations are the components of the direction vector, while the numbers that do not multiply t are the coordinates of the initial point. This makes it easy to write down parametric equations for a line through a given point in a given direction - for example, the line through (1, 0 -3) with direction vector [-2, 3, 0] has parametric equations

x = 1 + (-2)t y = 0 + 3t                 i.e.   z = -3 + 0t

x = 1 - 2t y = 3t z = -3 .

Conversely, the line with parametric equations

x = 5t
y = -4
z = 5 + 7t

has an initial point (0, -4, 5) and a direction vector [5, 0, 7].

For lines in 2-space, the set-up is identical to that in 3-space, with one coordinate less: the line through a point (x0, y0) with direction vector
[a, b] has parametric equations

x = x0 + at
y = y0 + bt.

So, for example, the line through (1, 2) with direction vector [3, 2] has parametric equations

x = 1 + 3t
y = 2 + 2t,

and the line with parametric equations

x = 3 y = 2 - t

i.e.

x = 3 + 0t y = 2 +(–1)t

is a vertical line through (3, 2).

How would you write the line y = mx + b in parametric form? (There are many different ways.)