Direction of a line in space

A line in space has a direction. We describe it by the angles α, β, γ it makes with the x, y and z axes.

Their cosines are the direction cosines: l = cos α, m = cos β, n = cos γ. They always satisfy l² + m² + n² = 1.

Direction cosines are often messy, so we also use direction ratios a, b, c: any three numbers in the same proportion as l, m, n. For example 1, 2, 2 and 2, 4, 4 are direction ratios of the same line.

Ratios to cosines: divide each ratio by √(a² + b² + c²). For 1, 2, 2: √(1 + 4 + 4) = 3, so l, m, n = 1/3, 2/3, 2/3. (A line has two opposite senses, so –1/3, –2/3, –2/3 is also valid.)

Line through two points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂): the direction ratios are the differences, x₂ – x₁, y₂ – y₁, z₂ – z₁.

Example: P(1, 2, 3), Q(3, 5, 9). Direction ratios 2, 3, 6. Length √(4 + 9 + 36) = 7, so the direction cosines are 2/7, 3/7, 6/7.

Easy cases: the x-axis has direction cosines 1, 0, 0; the y-axis 0, 1, 0; the z-axis 0, 0, 1.

Collinear points: A, B, C lie on one line if the direction ratios of AB and BC are proportional.

Remember the difference: direction cosines must satisfy l² + m² + n² = 1; direction ratios need not.