Q1. Find the direction cosines of a line with direction ratios (1, 2, 2).
Magnitude = √(1²+2²+2²) = √9 = 3
Direction cosines = (1/3, 2/3, 2/3)
Check: (1/3)² + (2/3)² + (2/3)² = 1/9+4/9+4/9 = 1 ✓
Three Dimensional Geometry — Class 12
Every line in space has a direction you can measure with three numbers. Compute direction cosines and the angle between two lines instantly.
Direction ratios & cosinesDirection ratios (a,b,c) describe a line's direction; direction cosines (l,m,n) are the same, normalised so l²+m²+n² = 1.
Equation of a lineThrough a point (x₁,y₁,z₁) with direction ratios (a,b,c): (x−x₁)/a = (y−y₁)/b = (z−z₁)/c.
Angle between two linescosθ = (a₁a₂+b₁b₂+c₁c₂) / (√(a₁²+b₁²+c₁²)·√(a₂²+b₂²+c₂²)), using the direction ratios of each line.
Angle between two lines calculator Interactive
Enter the direction ratios of two lines to find the angle between them.
Angle θ (degrees)—
If the result is 90°, the lines are perpendicular — that happens exactly when a₁a₂+b₁b₂+c₁c₂ = 0.
Solved examples
Formula sheet
| Quantity | Formula |
|---|---|
| Direction cosines relation | l² + m² + n² = 1 |
| Angle between two lines | cosθ = Σaᵢaⱼ / (|d₁||d₂|) |
| Perpendicular lines | a₁a₂+b₁b₂+c₁c₂ = 0 |
Common mistakes & exam wins
- Direction cosines are just direction ratios divided by the magnitude, so they always satisfy l²+m²+n²=1.
- Two lines are perpendicular exactly when a₁a₂+b₁b₂+c₁c₂ = 0.
- The equation of a line needs one point on it PLUS its direction ratios.
Frequently asked questions
What are direction cosines?
The cosines of the angles a line makes with the x, y and z axes; they satisfy l² + m² + n² = 1.
How do you find the angle between two lines in 3D?
Use cosθ = (a₁a₂+b₁b₂+c₁c₂)/(|d₁||d₂|), from the direction ratios of each line.
What is the condition for two lines to be perpendicular?
The dot product of their direction ratios is zero: a₁a₂+b₁b₂+c₁c₂ = 0.