NCERT Book Class 12 Maths Chapter 11 Three Dimensional Geometry

THREE DIMENSIONAL GEOMETRY 377

Chapter 11
THREE DIMENSIONAL GEOMETRY

v The moving power of mathematical invention is not
reasoning but imagination. – A. DEMORGAN v

11.1 Introduction
In Class XI, while studying Analytical Geometry in two
dimensions, and the introduction to three dimensional
geometry, we confined to the Cartesian methods only. In
the previous chapter of this book, we have studied some
basic concepts of vectors. We will now use vector algebra
to three dimensional geometry. The purpose of this
approach to 3-dimensional geometry is that it makes the
study simple and elegant*.
In this chapter, we shall study the direction cosines
and direction ratios of a line joining two points and also
discuss about the equations of lines and planes in space
under different conditions, angle between two lines, two
Leonhard Euler
planes, a line and a plane, shortest distance between two (1707-1783)
skew lines and distance of a point from a plane. Most of
the above results are obtained in vector form. Nevertheless, we shall also translate
these results in the Cartesian form which, at times, presents a more clear geometric
and analytic picture of the situation.
11.2 Direction Cosines and Direction Ratios of a Line
From Chapter 10, recall that if a directed line L passing through the origin makes
angles α, β and γ with x, y and z-axes, respectively, called direction angles, then cosine
of these angles, namely, cos α, cos β and cos γ are called direction cosines of the
directed line L.
If we reverse the direction of L, then the direction angles are replaced by their supplements,
i.e., , and . Thus, the signs of the direction cosines are reversed.
* For various activities in three dimensional geometry, one may refer to the Book
“A Hand Book for designing Mathematics Laboratory in Schools”, NCERT, 2005