Introduction

11.1 Introduction

The study of geometry in three dimensions began in Class XI, where we used Cartesian coordinates to locate points in space. That approach, while powerful, often made the algebra heavy and the geometry less transparent. Now, with the vector algebra developed in the previous chapter, we have a more elegant tool. Vectors capture both direction and magnitude in a single object, and their operations — addition, scalar multiplication, dot product, cross product — translate directly into geometric facts about lines and planes.

This chapter uses vector methods as the primary language, then translates each result into Cartesian form. The Cartesian form is not obsolete; it often gives a clearer geometric picture and is the form used in many examination problems. The key is fluency in both.

What This Chapter Covers

All topics are developed in both vector and Cartesian forms:

• Direction cosines and direction ratios of a line, and how to find them for a line joining two points.
• Equations of a line in space, under different given conditions (point and direction, two points).
• Angle between two lines — using direction cosines and direction ratios.
• Equations of a plane in space, under various conditions (normal vector and point, three points, two lines, line and point).
• Angle between two planes, and angle between a line and a plane.
• Shortest distance between two skew lines.
• Distance of a point from a plane.