Equation of a line

A line is fixed by one point on it and its direction. Both forms of the equation just record these two facts.

Vector form: r = a + λb. Here a is the position vector of a known point, b is a vector along the line, and λ is a number that slides along the line: each value of λ gives one point.

Cartesian form: (x – x₁)/a = (y – y₁)/b = (z – z₁)/c, where (x₁, y₁, z₁) is the point and a, b, c are direction ratios.

Example: the line through (1, –1, 2) and (3, 2, 0). Direction: 3 – 1, 2 – (–1), 0 – 2 = 2, 3, –2.

• Vector: r = î – ĵ + 2k̂ + λ(2î + 3ĵ – 2k̂).
• Cartesian: (x – 1)/2 = (y + 1)/3 = (z – 2)/(–2).

Reading a Cartesian equation: the point comes from the numbers subtracted on top (watch signs: y + 1 means y – (–1)); the direction comes from the bottoms.

Careful: each top must be written as (x – something), with x, y, z having coefficient +1. If you see (2 – y)/4, rewrite it as (y – 2)/(–4) first: the direction ratio is –4, not 4.

Parallel lines have the same (or proportional) direction ratios. So a line through a new point parallel to a given line keeps the given bottoms and changes the tops.

A line perpendicular to two given lines has direction b₁ × b₂, the cross product of their directions (Chapter 10).