a line in space

In the plane, a single equation like y = 2x + 1 captures a line. In three dimensions one equation usually describes a whole surface, not a line, so we need a richer description. The cleanest is to say: start at a known point and walk in a fixed direction. That single sentence — a point plus a direction — pins down a line in space exactly.

Let the line pass through the point A with position vector a, and run in the direction of the vector d (its direction vector). Every point on the line is reached from A by going some multiple t of d, so the vector equation is r = a + t d, where the parameter t sweeps over all real numbers. Splitting into components with a = (x_0, y_0, z_0) and d = (a, b, c) gives the parametric equations x = x_0 + t a, y = y_0 + t b, z = z_0 + t c. Eliminating t (when a, b, c are all nonzero) gives the symmetric form (x - x_0)/a = (y - y_0)/b = (z - z_0)/c, which shows the direction ratios in the denominators. Two lines are parallel when their direction vectors are scalar multiples, and perpendicular when their direction vectors have dot product zero.

This point-and-direction description powers everything about lines in space: finding where a line meets a plane, the angle between two lines (from the dot product of their directions), and distances. A subtlety to respect: the same line has infinitely many valid equations — you may start from any point on it and use any nonzero scalar multiple of its direction. So two different-looking equations can describe the identical line; to check, see whether the directions are parallel and a point of one lies on the other. And if a direction component is zero, the symmetric form breaks for that coordinate (you cannot divide by zero) — handle that axis separately, e.g. as the plain statement that the coordinate is constant.