8 min readConcept sheet1-mark MCQ and a 2-mark question

Direction cosines and direction ratios of a line

A line makes angles α, β, γ with the x, y and z-axes. Their cosines l, m, n are its direction cosines, and l² + m² + n² = 1.

Direction cosines (DCs)l = cos α, m = cos β, n = cos γ, where α, β, γ are the angles the directed line makes with the positive x, y and z-axes

Key identityl² + m² + n² = 1

Direction ratios (DRs)Any three numbers a, b, c proportional to l, m, n: a = kl, b = km, c = kn for some k ≠ 0

From DRs to DCsl = ±a/√(a² + b² + c²), m = ±b/√(a² + b² + c²), n = ±c/√(a² + b² + c²), with the same sign for all three

What direction cosines are

Think of a line in space with a direction marked on it (a directed line). It makes three angles with the positive x, y and z-axes: α, β and γ. These are its direction angles. Their cosines, l = cos α, m = cos β, n = cos γ, are the direction cosines of the line. If you reverse the direction, each angle becomes its supplement (π − α and so on). So the direction cosines become −l, −m, −n. A line with no marked direction therefore has two sets of direction cosines. A line that does not pass through the origin has the same direction cosines as the parallel line through the origin.

Proof that l² + m² + n² = 1

Take the line through the origin O and a point P(x, y, z) on it, with OP = r.

1
Drop a perpendicular from P to the x-axis. In the right triangle formed, cos α = x/r, so x = lr.
2
In the same way, y = mr and z = nr.
3
By the distance formula, OP² = x² + y² + z², so r² = x² + y² + z².
4
Substitute: r² = l²r² + m²r² + n²r² = r²(l² + m² + n²).
5
Divide both sides by r² (r ≠ 0): l² + m² + n² = 1.
6★
So, the squares of the direction cosines of any line add up to 1.

★ key step

Direction cosines vs direction ratios

Direction cosines l, m, nUnique for a directed line (up to sign for an undirected line). They always satisfy l² + m² + n² = 1, and each lies between −1 and 1. Example: the x-axis has DCs 1, 0, 0.

Direction ratios a, b, cInfinitely many sets: any non-zero multiple works. Their squares need not add up to 1. Example: 1, 2, 2 and 2, 4, 4 are DRs of the same line.

DCs of the axes: x-axis 1, 0, 0; y-axis 0, 1, 0; z-axis 0, 0, 1.

Example 1: from direction ratios to direction cosines

Problem

Find the direction cosines of a line whose direction ratios are 4, −4, 2.

Working

Formula: l = a/√(a² + b² + c²), m = b/√(a² + b² + c²), n = c/√(a² + b² + c²).
√(a² + b² + c²) = √(4² + (−4)² + 2²) = √(16 + 16 + 4) = √36 = 6
l = 4/6 = 2/3, m = −4/6 = −2/3, n = 2/6 = 1/3
Check: 4/9 + 4/9 + 1/9 = 9/9 = 1

Answer

The direction cosines are 2/3, −2/3, 1/3 (or −2/3, 2/3, −1/3 for the opposite direction).

Example 2: a line equally inclined to the axes

Problem

A line is equally inclined to all three coordinate axes. Find its direction cosines.

Working

Let each angle be α. Then l = m = n = cos α.
Use l² + m² + n² = 1: 3 cos²α = 1.
cos²α = 1/3, so cos α = ±1/√3.

Answer

The direction cosines are ±1/√3, ±1/√3, ±1/√3 (all with the same sign).

Example 3: can these be the angles?

Problem

Can a line make angles of 45°, 60° and 30° with the x, y and z-axes?

Working

If it could, cos²45° + cos²60° + cos²30° would equal 1.
cos²45° = 1/2, cos²60° = 1/4, cos²30° = 3/4
Sum = 2/4 + 1/4 + 3/4 = 6/4 = 3/2
3/2 ≠ 1, so the identity fails.

Answer

No. No line can make these three angles with the axes.

In real life

Flight software for drones and satellites describes a direction in space by three numbers like l, m, n. A drone climbing in a straight line keeps the same direction cosines all along its path.

Exam angle

1 marks: “A line makes angles 90°, 150° and θ with the x, y and z-axes, where θ is acute. Find θ.” Use l² + m² + n² = 1: cos²90° + cos²150° + cos²θ = 1, so 0 + 3/4 + cos²θ = 1. cos²θ = 1/4, cos θ = 1/2 (θ is acute), so θ = 60°.

2 marks: “If a line makes angles α, β, γ with the coordinate axes, prove that sin²α + sin²β + sin²γ = 2.” sin²α + sin²β + sin²γ = (1 − cos²α) + (1 − cos²β) + (1 − cos²γ) = 3 − (l² + m² + n²) = 3 − 1 = 2.

2 marks: “Find the direction cosines of a line whose direction ratios are 1, −2, 2.” √(1 + 4 + 4) = 3. DCs: 1/3, −2/3, 2/3 (or −1/3, 2/3, −2/3).

Remember it

Cosines square to one: l² + m² + n² = 1. DRs are any multiple; divide by √(a² + b² + c²) to get the DCs.