Equation of a line

A line is fixed by a point and a direction, or by two points.

Vector form: the line through the point with position vector a, parallel to b, is r = a + λb. Each real λ gives one point of the line, and every point comes from one λ.

Writing b = aî + bĵ + ck̂, the three numbers a, b, c serve as direction ratios. Here b is a direction, not a length; do not confuse it with |b|.

Parametric form: through (x₁, y₁, z₁) with ratios a, b, c, the points are x = x₁ + λa, y = y₁ + λb, z = z₁ + λc.

Cartesian form: eliminating λ gives (x − x₁)/a = (y − y₁)/b = (z − z₁)/c. With direction cosines in place of ratios it reads (x − x₁)/l = (y − y₁)/m = (z − z₁)/n.

Through two points with position vectors a and b the line is r = a + λ(b − a).

Worked: through (5, 2, −4) parallel to 3î + 2ĵ − 8k̂ the line is r = 5î + 2ĵ − 4k̂ + λ(3î + 2ĵ − 8k̂), that is (x − 5)/3 = (y − 2)/2 = (z + 4)/(−8).

Read the coefficients before quoting ratios: (1 − x)/3 = (x − 1)/(−3) has ratio −3 for x, and (7 − 7x)/3 = (x − 1)/(−3/7).