Lesson 1 of 9 · 7 min

Direction cosines and direction ratios

NCERT §11.1–11.2

For the annual day the lighting crew rigs lasers in the school hall. The first sits on the floor at F(1, 2, 0) and hits a mirror on the ceiling at C(3, 5, 6). How do you tell someone which way it points?

The story this chapter follows: The school hall laser show

For the annual day the lighting crew rigs laser beams across the school hall. With a floor corner as origin, x across the hall, y along it and z up to the ceiling at 6 m, every beam is a line in space, and the chapter follows the crew as they aim, check and space their lasers. All numbers are made up for the story.

The lesson in notes

In short

A directed line through the origin makes angles α, β, γ with the positive x-, y- and z-axes (its direction angles). l = cos α, m = cos β, n = cos γ are its direction cosines.

Reversing the line replaces each angle by its supplement, so every direction cosine changes sign. A line that is not directed therefore has two sets, ±(l, m, n); fixing a direction makes the set unique.

A line not through the origin has the direction cosines of the parallel line through the origin: parallel lines share them.

l² + m² + n² = 1 always.

Direction ratios are any three numbers a, b, c proportional to l, m, n: a = λl, b = λm, c = λn for some non-zero λ. A line has infinitely many sets of direction ratios, and any two of them are proportional.

From ratios to cosines: l = ±a/√(a² + b² + c²), m = ±b/√(a² + b² + c²), n = ±c/√(a² + b² + c²), with the same sign taken throughout.

Worked: angles 90°, 60°, 30° with the axes give direction cosines 0, 1/2, √3/2. Ratios 2, −1, −2 give 2/3, −1/3, −2/3, since √(4 + 1 + 4) = 3.

The axes themselves: the x-axis has direction cosines 1, 0, 0; the y-axis 0, 1, 0; the z-axis 0, 0, 1.