Class 12 Maths Chapter 11: Three Dimensional Geometry

Three Dimensional Geometry is Chapter 11 of NCERT Class 12 Maths. Building on vector algebra, it studies straight lines in space: their direction cosines and direction ratios, their vector and Cartesian equations, the angle between two lines, and the shortest distance between them. Watch the two video lessons below, then use these notes and solved examples to revise.

What you will learn

Direction cosines and direction ratios of a line and how they are related
The equation of a line through a point parallel to a vector, in vector and Cartesian form
The equation of a line through two given points
The angle between two lines and the conditions for parallel and perpendicular lines
The shortest distance between skew lines and between parallel lines

Video Lessons

Three Dimensional Geometry - Part 1

Three Dimensional Geometry - Part 2

Key Concepts and Formulas

Direction Cosines of a Line

If a directed line makes angles α, β and γ with the positive x, y and z axes, the cosines of these angles are its direction cosines. Reversing the direction of the line changes the sign of all three.

l = cos α, m = cos β, n = cos γ
l² + m² + n² = 1
Line joining P(x₁, y₁, z₁) and Q(x₂, y₂, z₂):
l = (x₂ − x₁)/PQ, m = (y₂ − y₁)/PQ,
n = (z₂ − z₁)/PQ

Direction angles α, β and γ of a line OP with the coordinate axes, Class 12 Three Dimensional Geometry — l = cos α, m = cos β, n = cos γ

Direction Ratios

Any three numbers proportional to the direction cosines are called direction ratios. A line has infinitely many sets of direction ratios but only two sets of direction cosines.

If a, b, c are direction ratios:
l = ±a/√(a² + b² + c²), similarly m and n
Through two points: x₂ − x₁, y₂ − y₁, z₂ − z₁

Three points are collinear if the direction ratios of the lines joining them are proportional

Equation of a Line in Space

A line is fixed by one point on it and its direction, or by two points on it. Learn both the vector form and the Cartesian (symmetric) form; in the Cartesian form, the denominators give the direction ratios.

Point a, parallel to b: r = a + λb
Through (x₁, y₁, z₁) with d.r. a, b, c:
(x − x₁)/a = (y − y₁)/b = (z − z₁)/c
Through two points a and b: r = a + λ(b − a)
(x − x₁)/(x₂ − x₁) = (y − y₁)/(y₂ − y₁)
                   = (z − z₁)/(z₂ − z₁)

Angle Between Two Lines

The angle between two lines is the angle between their direction vectors. Taking the modulus gives the acute angle.

cos θ = |b₁ · b₂|/(|b₁||b₂|)
With d.r. a₁, b₁, c₁ and a₂, b₂, c₂:
cos θ = |a₁a₂ + b₁b₂ + c₁c₂|
        / (√(a₁² + b₁² + c₁²) √(a₂² + b₂² + c₂²))
Perpendicular: a₁a₂ + b₁b₂ + c₁c₂ = 0
Parallel: a₁/a₂ = b₁/b₂ = c₁/c₂

Shortest Distance Between Two Lines

Lines in space that are neither parallel nor intersecting are called skew lines. The shortest distance is measured along the common perpendicular.

Skew: r = a₁ + λb₁ and r = a₂ + μb₂
d = |(b₁ × b₂) · (a₂ − a₁)| / |b₁ × b₂|
Parallel: r = a₁ + λb and r = a₂ + μb
d = |b × (a₂ − a₁)| / |b|

Shortest distance between two skew lines along their common perpendicular, Class 12 Maths Three Dimensional Geometry — The shortest distance runs along the common perpendicular

If the skew line distance comes out 0, the lines intersect

Solved Examples

Example 1: Find the direction cosines of a line with direction ratios 2, −6, 3.

√(2² + (−6)² + 3²) = √(4 + 36 + 9) = √49 = 7
∴ Direction cosines are 2/7, −6/7, 3/7

Example 2: Can 1/2, 1/2, 1/2 be the direction cosines of a line?

l² + m² + n² = 1/4 + 1/4 + 1/4 = 3/4
3/4 ≠ 1
∴ No, they cannot be direction cosines

Example 3: Find the equation of the line through (1, −2, 4) parallel to 3î + ĵ − 2k̂.

a = î − 2ĵ + 4k̂, b = 3î + ĵ − 2k̂
Vector form: r = (î − 2ĵ + 4k̂) + λ(3î + ĵ − 2k̂)
∴ (x − 1)/3 = (y + 2)/1 = (z − 4)/(−2)

Example 4: Find the equation of the line through A(2, 1, −3) and B(5, −1, 1).

Direction ratios: 5 − 2, −1 − 1, 1 − (−3)
                = 3, −2, 4
∴ (x − 2)/3 = (y − 1)/(−2) = (z + 3)/4

Example 5: Find the angle between r = (2î − k̂) + λ(î + ĵ) and r = (î + 3ĵ) + μ(ĵ + k̂).

b₁ = î + ĵ, b₂ = ĵ + k̂
b₁ · b₂ = 0 + 1 + 0 = 1
|b₁| = √2, |b₂| = √2
cos θ = 1/(√2 × √2) = 1/2
∴ θ = 60°

Example 6: Find the shortest distance between r = (î + 2ĵ) + λ(î − ĵ + k̂) and r = (2î + 3k̂) + μ(2î + ĵ − k̂).

b₁ × b₂ = (1 − 1)î − (−1 − 2)ĵ + (1 + 2)k̂
b₁ × b₂ = 3ĵ + 3k̂, |b₁ × b₂| = 3√2
a₂ − a₁ = î − 2ĵ + 3k̂
(b₁ × b₂) · (a₂ − a₁) = 0 − 6 + 9 = 3
d = 3/(3√2)
∴ d = 1/√2 units

Example 7: Find the distance between the parallel lines r = (î + ĵ) + λ(2î − ĵ + 2k̂) and r = (2î + ĵ − k̂) + μ(2î − ĵ + 2k̂).

b = 2î − ĵ + 2k̂, |b| = 3
a₂ − a₁ = î − k̂
b × (a₂ − a₁) = î + 4ĵ + k̂
|b × (a₂ − a₁)| = √18 = 3√2
d = 3√2/3
∴ d = √2 units

Common Mistakes to Avoid

Before reading direction ratios from a Cartesian equation, make the coefficients of x, y and z equal to 1; for example, (3 − x)/2 means direction ratio −2, not 2.
Use the modulus in the angle and distance formulas, because an angle between lines and a distance cannot be negative.
Check whether the lines are parallel first; the skew line formula fails when b₁ × b₂ = 0.
Do not mix up direction ratios (any proportional numbers) with direction cosines (which must satisfy l² + m² + n² = 1).

Quick Revision

l² + m² + n² = 1
Line: r = a + λb
(x − x₁)/a = (y − y₁)/b = (z − z₁)/c
cos θ = |b₁ · b₂|/(|b₁||b₂|)
Perpendicular: a₁a₂ + b₁b₂ + c₁c₂ = 0
Skew: d = |(b₁ × b₂) · (a₂ − a₁)|/|b₁ × b₂|
Parallel: d = |b × (a₂ − a₁)|/|b|