Equations of Lines in 3D

Calculus 3 · Axiom Academy

LESSON Equations of Lines in 3D A line in space is one point plus one direction: watch trace out every point as t runs. 1. One Point, One Direction, a Whole Line Start at a known point P_0 with position vector . Add a scaled copy of the direction vector . As the parameter t slides, the tip of walks along the line — forward for t > 0 , backward for t < 0 , and sitting exactly on P_0 at t = 0 . A point P_0 says where to start A direction says which way to go 2. Three Forms of the Same Line Reading off the three coordinates of gives the parametric form. Solving each of those for t and equating them removes the parameter — that is the symmetric form. All three describe the identical set of points. — compact; best for proofs and computer graphics. Equate the three t -values — best for testing whether a point lies on the line. If, say, a = 0 then x = x_0 is fixed; write the symmetric form with the remaining coordinates only. Each parametric equation is solved for the same parameter t . Setting those equal is just saying "the same t moved all three coordinates" — so the symmetric chain and the vector equation are two views of one motion. Take the concrete line through P_0 = (1, 2, 3) with direction . Every t produces one point on the line, and a point is on the line exactly when its three symmetric ratios all match. Generate a point: at t = 1 , the coordinates are — a point guaranteed to sit on the line.

This is the written version of the interactive lesson above. See the full Calculus 3 course.