NCERT Class 12 Maths Three Dimensional Geometry

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 planes, a line and a plane, shortest distance between two 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 andγ with x, y and z-axes, respectively, called direction angles, then cosine of these angles, namely, cosα, 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.,∏ −α , ∏ − β and∏ − γ. Thus, the signs of the direction cosines are reversed.

Please refer to attached file for NCERT Class 12 Maths Three Dimensional Geometry