NCERT Solutions for Class 12 Mathematics Chapter 11: Three-Dimensional Geometry – Free PDF Download

Chapter 11 of Class 12 Maths is Three-Dimensional Geometry. It extends vector concepts to lines and planes in 3D space, covering direction cosines/ratios, equations of lines and planes, and distances/angles between them. These Class 12 Mathematics Chapter 11 solutions are also useful as quick revision notes before exams.

Direction Cosines and Direction Ratios

If a line makes angles α,β,γ with the x,y,z axes, cosα,cosβ,cosγ are its direction cosines (l,m,n), satisfying l²+m²+n²=1. Any numbers proportional to l,m,n are direction ratios.

Equation of a Line in Space

Vector form: r=a+λb (through point with position vector a, along direction b). Cartesian form: (x−x₁)/a=(y−y₁)/b=(z−z₁)/c.

Angle Between Two Lines

Using direction ratios: cosθ=|a₁a₂+b₁b₂+c₁c₂|/[√(a₁²+b₁²+c₁²)√(a₂²+b₂²+c₂²)]. Lines are perpendicular if a₁a₂+b₁b₂+c₁c₂=0.

Equation of a Plane

Vector form (normal form): r·n=d. Cartesian form: ax+by+cz=d, where a,b,c are direction ratios of the normal.

Distances

Distance of a point from a plane: |ax₁+by₁+cz₁−d|/√(a²+b²+c²). Shortest distance between skew lines is computed via the scalar triple product formula.

Frequently Asked Questions

What is the relation between direction cosines?
l²+m²+n²=1.

How do you check if two lines are perpendicular using direction ratios?
Their dot product (a₁a₂+b₁b₂+c₁c₂) equals zero.

What is the formula for distance of a point from a plane?
|ax₁+by₁+cz₁−d|/√(a²+b²+c²).

Written by Satish

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