medium difficulty9 min

Direction Cosines and Direction Ratios

Direction cosines of a line are the cosines of the angles made by the line with the positive x-, y-, and z-axes, usually denoted by l, m, and n. Direction ratios are any three numbers proportional to the direction cosines.

In one line: Ratios show direction; cosines are ratios made unit length.

Practice This Concept

Understand it

If a line makes angles α, β, and γ with the coordinate axes, then l = cos α, m = cos β, and n = cos γ. Since these come from a unit direction vector, they always satisfy l² + m² + n² = 1. Direction ratios a, b, c only show direction proportion, so they need not satisfy this identity until converted into direction cosines.

Example

For direction ratios 2, -1, 2, the magnitude factor is √(2² + (-1)² + 2²) = 3. The direction cosines are 2/3, -1/3, 2/3.

Formula and rule

Key rule: if l, m, n are direction cosines, then l² + m² + n² = 1.

If a, b, c are direction ratios, then direction cosines are ±a/√(a²+b²+c²), ±b/√(a²+b²+c²), ±c/√(a²+b²+c²), with sign chosen according to the line's direction.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Given direction ratios 3, 4, 12, find direction cosines.

1Compute √(3² + 4² + 12²) = √(9 + 16 + 144) = √169 = 13.
2Therefore l = 3/13, m = 4/13, n = 12/13.
3Check: 9/169 + 16/169 + 144/169 = 169/169 = 1.

Method you can reuse

1Identify whether the given numbers are ratios or cosines.
2For ratios, calculate √(a²+b²+c²).
3Divide each ratio by this value.
4For cosines, check the identity l² + m² + n² = 1.
5If one cosine is missing, substitute the known values and solve carefully with the correct sign condition if given.

Visual tip

A simple 3D axes sketch may help show the angles α, β, γ with the coordinate axes, but most exam questions can be solved using the formula directly.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Find the direction cosines of a line whose direction ratios are 1, 2, 2.

Think first, then show answer

Magnitude factor = √(1² + 2² + 2²) = 3, so the direction cosines are 1/3, 2/3, 2/3.

Common wrong answer

Writing 3, 4, 12 as direction cosines without normalising, which fails because 3² + 4² + 12² is not 1.

Don't mix this up

Students often treat direction ratios as direction cosines and wrongly expect a² + b² + c² = 1 for ratios.

Exam tip

When the question gives direction ratios, first divide by √(a² + b² + c²) if direction cosines are required.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Usually asked as a short answer problem: convert direction ratios to direction cosines, verify whether given numbers can be direction cosines, or find a missing direction cosine using l² + m² + n² = 1.

1
Enough for 1 mark
Direction cosines of a line are the cosines of the angles made by the line with the positive x-, y-, and z-axes, usually denoted by l, m, and n. Direction ratios are any three numbers proportional to the direction cosines.
2
Add for 2 marks
Example: For direction ratios 2, -1, 2, the magnitude factor is √(2² + (-1)² + 2²) = 3. The direction cosines are 2/3, -1/3, 2/3.
3
Add for 3 marks
If a line makes angles α, β, and γ with the coordinate axes, then l = cos α, m = cos β, and n = cos γ.

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