There is a standard physical interpretation for the direction vector of a line.

Suppose we have a line with vector equation r = r0 + td. If t represents time, measured in some appropriate units (seconds, say), then for each value of t, the point with position vector r is at a new position, i.e. it moves along the line. It's always moving parallel to the direction vector, and we can ask: how fast is it moving?

Between any time t and time t+1, it moves from position r0 + td to
r0 + (t+1)d. Its displacement during this time is thus

{r0 + (t+1)d} - {r0 + td} = d.

It covers this displacement in a time (t + 1) - t = 1 time unit. We thus have a displacement of size/direction d for every unit of time, i.e. the velocity of the point is d.

To summarize: if r = r0 + td describes the position at time t of a point moving along a line, then the direction vector d may be interpreted as the velocity vector of the point.

In this diagram, the point moves along the line with a velocity determined by the direction vector (time is measured in seconds). You can adjust the velocity by dragging the direction vector, and you can change the starting position of the point.