Three Dimensional Geometry
Learn Three Dimensional Geometry with notes, examples, and practice questions.
Chapter Overview
Three-dimensional geometry extends coordinate geometry to space. Lines and planes are described using vectors and Cartesian equations — essential for JEE and board exams.
Direction Cosines and Ratios
D.C.
l = cos α, m = cos β, n = cos γ
Relation
l² + m² + n² = 1
D.R.
a,b,c proportional to l,m,n → l=a/√(a²+b²+c²)
Line in 3D
Vector form
r⃗ = a⃗ + λb⃗ (point a⃗, parallel to b⃗)
Cartesian (symmetric)
(x−x₁)/a = (y−y₁)/b = (z−z₁)/c
Through two points
Direction = (x₂−x₁, y₂−y₁, z₂−z₁)
Distance point to line
Use cross product formula
Plane in 3D
Normal form
r⃗·n̂ = d (n̂ unit normal)
Cartesian
ax + by + cz + d = 0; normal (a,b,c)
Through point + normal
a(x−x₁)+b(y−y₁)+c(z−z₁)=0
Distance origin
|d|/√(a²+b²+c²)
Angle between planes
cos θ = |n₁·n₂|/(|n₁||n₂|)
Key Distances
- Distance between parallel planes: |d₁−d₂|/|n|
- Skew lines: shortest distance via scalar triple product
- Coplanar lines: intersect or parallel
Solved Examples
Step-by-step solutions — read each step before checking the final answer.
Example 1: Line Equation
Line through (1,2,3) parallel to (2,−1,1). Write Cartesian form.
- 1(x−1)/2 = (y−2)/(−1) = (z−3)/1
Answer
(x−1)/2 = (y−2)/(−1) = (z−3)/1
Example 2: Plane Equation
Plane through (1,0,0) with normal (1,1,1).
- 11(x−1)+1(y−0)+1(z−0)=0
- 2x+y+z=1
Answer
x + y + z = 1
Example 3: Distance from Origin
Distance from origin to plane 2x − y + 2z + 9 = 0.
- 1|9|/√(4+1+4) = 9/3 = 3
Answer
3 units
Key Points to Remember
- ✓ Line needs point + direction; plane needs point + normal
- ✓ Convert between vector and Cartesian forms fluently
- ✓ Angle between lines uses direction vectors
Exam Tips
- • Normalise only when formula requires unit vector
- • For distance problems, identify correct formula (point-plane vs skew lines)
- • Check if lines intersect by solving parametric equations
Formula Cheat Sheet
Quick reference — NCERT board exam formulas
3D Geometry
- Direction cosines:l² + m² + n² = 1
- Angle between lines:cos θ = |l₁l₂ + m₁m₂ + n₁n₂|
- Distance:PQ = √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²]
- Line: (x−x₁)/a = (y−y₁)/b = (z−z₁)/c
- Plane: ax + by + cz + d = 0
Practice MCQs — Three Dimensional Geometry
Test your understanding with topic-wise multiple choice questions. Explanations appear after each answer.
l²+m²+n² equals: