Vector and Cartesian equations of a line

A straight line extends infinitely in two opposite directions, so no single equation of the kind we meet with curves in a plane can describe it in three dimensions directly the way $y = mx + c$ describes a line in a plane. Instead, we describe a line in space by picking out every point that lies on it through its position vector. If we fix an origin O, then every point in space corresponds to exactly one position vector from O, and a line becomes the set of all position vectors of the points lying on it. Throughout this chapter, the position vector of a fixed, known point is written with a subscript-free symbol such as $\bar{a}$ or $\bar{b}$, while the position vector of a point that is allowed to move along the line — a general point on the line — is always written as $\bar{r}$. Because a line is one-dimensional even though it sits inside three-dimensional space, its points can be listed using a single real-number parameter; once we know how $\bar{r}$ depends on that parameter, we know the whole line. The next two subsections …