Class 12 Mathematics Chapter 11: Three-Dimensional Geometry
CBSE 2026–27 | Detailed NCERT-aligned free resource
Three-Dimensional Geometry extends coordinate geometry into space. The chapter develops direction ratios and cosines, equations of lines, angles between lines, shortest distance, and the relationship between lines.
1. Coordinates in Space
A point in three-dimensional space is represented by (x,y,z). The coordinate axes are mutually perpendicular.
│ ╲
│ ╲ P(x,y,z)
│ ╲
└────────→ x
╲
y
2. Direction Ratios and Direction Cosines
If a line has direction ratios a,b,c, its direction cosines are proportional to them. For direction cosines l,m,n: l²+m²+n²=1.
Example
For direction ratios 2,−3,6, the magnitude is √(4+9+36)=7. Direction cosines are 2/7, −3/7 and 6/7.
3. Equation of a Line
A line through point (x₁,y₁,z₁) with direction ratios a,b,c can be written in symmetric form as (x−x₁)/a=(y−y₁)/b=(z−z₁)/c.
In vector form, r=a+λb, where a is the position vector of a point on the line and b is a direction vector.
4. Angle Between Two Lines
If the direction ratios are (a₁,b₁,c₁) and (a₂,b₂,c₂), then cosθ=(a₁a₂+b₁b₂+c₁c₂)/(√(a₁²+b₁²+c₁²)√(a₂²+b₂²+c₂²)).
5. Intersecting, Parallel and Skew Lines
| Type | Meaning |
|---|---|
| Parallel | Same or proportional direction vectors and no intersection |
| Intersecting | Meet at one point |
| Skew | Neither parallel nor intersecting; not coplanar |
6. Shortest Distance Between Skew Lines
For r=a₁+λb₁ and r=a₂+μb₂, the shortest distance between skew lines is |(a₂−a₁)·(b₁×b₂)|/|b₁×b₂|.
╲ ⟂ shortest segment
╲────────╲ Line 2
7. Distance Between Parallel Lines
For two parallel lines, use a point from one line and the direction vector of the lines. The distance can be obtained using |(a₂−a₁)×b|/|b|.
8. Worked Example: Angle
For direction vectors b₁=i+2j+2k and b₂=2i−j+k, their dot product is 2−2+2=2. Their magnitudes are 3 and √6. Therefore cosθ=2/(3√6).
9. Common Mistakes
- Confusing direction ratios with coordinates of a point.
- Forgetting that skew lines do not lie in the same plane.
- Using the angle formula without taking the acute angle when required.
- Mixing the point vector and direction vector in line equations.
10. Practice Set
- Find direction cosines from given direction ratios.
- Write the equation of a line through a point with a given direction.
- Find the angle between two lines.
- Determine whether two lines are parallel, intersecting or skew.
- Find the shortest distance between two skew lines.