Class 12 Mathematics Chapter 11 Three-Dimensional Geometry – Revision Notes

Three-Dimensional Geometry extends coordinate geometry into space, using direction cosines and ratios to describe lines and planes and to calculate distances between them.

Last Updated: September 23, 2026

Direction Cosines/Ratios

• l²+m²+n²=1.
• Parallel lines: a₁/a₂=b₁/b₂=c₁/c₂. Perpendicular: a₁a₂+b₁b₂+c₁c₂=0.

Lines & Planes

• Line: r=a+λb; Cartesian (x−x₁)/a=(y−y₁)/b=(z−z₁)/c.
• Plane: r·n=d; Cartesian ax+by+cz=d.

Distances

• Point to plane: |ax₁+by₁+cz₁−d|/√(a²+b²+c²).
• Skew lines: scalar triple product formula.

One-Line Summary

3D Geometry extends vectors to describe lines and planes in space using direction cosines/ratios, with distance and angle formulas built from dot and cross products.

Frequently Asked Questions

How is the equation of a line in three-dimensional space represented?
A line in 3D is represented using a point it passes through and its direction ratios or direction cosines, expressed either in vector form or in symmetric Cartesian form.

How is the angle between two planes determined in this chapter?
The angle between two planes is found using the angle between their normal vectors, calculated through the dot product formula, since the orientation of a plane is fully described by the direction its normal points in.

Written by Satish

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