# Three Dimensional Geometry, Class 12 Mathematics | SwaVid

Three Dimensional Geometry, NCERT Class 12 Mathematics: notes with key ideas, formulas, common mistakes and 3 quick checks with worked answers.

Canonical: https://www.swavid.com/cbse/class-12/maths/chapter/three-dimensional-geometry

Source: https://www.swavid.com/cbse/class-12/maths/chapter/three-dimensional-geometry

# Three Dimensional Geometry

## Formulas and definitions

## Key ideas

## Common mistakes

## Quick checks

## Keep going

## Questions parents ask

## Class 12 Mathematics on SwaVid

### Direction cosines and direction ratios

### Equation of a line through a given point

### Equation of a line through two points

### Skew lines and shortest distance

### Distance between parallel lines

### Next chapter

### Related chapter

### All chapters

SwaVid › CBSE › Class 12 › Mathematics › Three Dimensional Geometry

Notes written by SwaVid from the NCERT textbook.

Class 12 Mathematics, Chapter 11. Key ideas, formulas and common mistakes.

CBSE does not print marks for a single chapter.

Short answer. This chapter studies lines in three-dimensional space using vector and Cartesian methods. Students learn to determine direction cosines, equations of lines, and the shortest distance between skew or parallel lines.

Source: NCERT Mathematics Part II (Reprint 2026-27) , Chapter 11, Three Dimensional Geometry

If a directed line makes angles α, β, and γ with the positive coordinate axes, then cos α, cos β, and cos γ are its direction cosines, denoted by l, m, and n. Any three numbers proportional to the direction cosines are called direction ratios. The direction cosines satisfy l² + m² + n² = 1.

A straight line passing through a point with position vector a and parallel to a vector b has the vector equation r = a + λb. In Cartesian form, the line passing through (x₁, y₁, z₁) with direction ratios a, b, c is given by (x - x₁) / a = (y - y₁) / b = (z - z₁) / c.

A line passing through two points with position vectors a and b has the vector equation r = a + λ(b - a). Its Cartesian equation is (x - x₁) / (x₂ - x₁) = (y - y₁) / (y₂ - y₁) = (z - z₁) / (z₂ - z₁).

Lines in space that are neither parallel nor intersecting are called skew lines, and they lie in different planes. The shortest distance between two skew lines r = a₁ + λb₁ and r = a₂ + μb₂ is the absolute value of ((b₁ × b₂) · (a₂ - a₁)) / |b₁ × b₂|.

Two parallel lines have the same direction vector b and equations r = a₁ + λb and r = a₂ + μb. The shortest distance between them is given by |b × (a₂ - a₁)| / |b|.

Try each question first. Then tap it to see the answer.

Compute √(a² + b² + c²) = √(2² + (-1)² + (-2)²) = √(4 + 1 + 4) = √9 = 3

Divide each direction ratio by 3: l = 2/3, m = -1/3, n = -2/3

2/3, -1/3, -2/3

Identify (x₁, y₁, z₁) = (1, -2, 3) and direction ratios a = 2, b = 3, c = -4

Substitute into (x - x₁) / a = (y - y₁) / b = (z - z₁) / c

Write the equation: (x - 1) / 2 = (y + 2) / 3 = (z - 3) / (-4)

(x - 1) / 2 = (y + 2) / 3 = (z - 3) / (-4)

Here a₁ = i + 2j - 4k, a₂ = 3i + 3j - 5k, and b = 2i + 3j + 6k

a₂ - a₁ = (3 - 1)i + (3 - 2)j + (-5 - (-4))k = 2i + j - k

b × (a₂ - a₁) = |i j k; 2 3 6; 2 1 -1| = i(-3 - 6) - j(-2 - 12) + k(2 - 6) = -9i + 14j - 4k

|b × (a₂ - a₁)| = √((-9)² + 14² + (-4)²) = √(81 + 196 + 16) = √293

|b| = √(2² + 3² + 6²) = √(4 + 9 + 36) = √49 = 7

Distance d = |b × (a₂ - a₁)| / |b| = √293 / 7

√293 / 7

Linear Programming .

Vector Algebra .

The 13 chapters of Class 12 Mathematics .

CBSE does not give marks to one chapter. It prints marks for units or parts, and the chapter list page shows them all.

No. They are written by SwaVid from the NCERT textbook. The CBSE curriculum is the reference for what is assessed.

Linear Programming.

SwaVid is an AI tutor for Indian students in Classes 6 to 12. It finds the earlier ideas a child missed, then teaches today’s chapter. Start free to see how it works.

- Ch 11 NCERT book
- 80 marks, theory paper
- 3 quick checks
- Treating direction ratios as direction cosines directly in formulas . Direction ratios a, b, c do not satisfy a² + b² + c² = 1. Always divide each ratio by √(a² + b² + c²) to convert them into direction cosines.
- Reading direction ratios from non-standard Cartesian forms . Ensure the numerators have coefficients of +1 for x, y, and z, such as (x - x₁). If a term is written as (1 - x) / 2, rewrite it as (x - 1) / (-2) before noting the ratio.
- Using the skew lines formula for parallel lines . Check if b₁ and b₂ are scalar multiples of each other. If they are parallel, b₁ × b₂ is zero, so use the formula d = |b × (a₂ - a₁)| / |b| instead.

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