THREE DIMENSIONAL GEOMETRY - NCERT

THREE DIMENSIONAL GEOMETRY463 The moving power of mathematical invention is notreasoning but imagination. IntroductionIn Class XI, while studying Analytical GEOMETRY in twodimensions, and the introduction to THREE dimensionalgeometry, we confined to the Cartesian methods only. Inthe previous chapter of this book, we have studied somebasic concepts of vectors. We will now use vector algebrato THREE DIMENSIONAL GEOMETRY . The purpose of thisapproach to 3- DIMENSIONAL GEOMETRY is that it makes thestudy simple and elegant*.In this chapter, we shall study the direction cosinesand direction ratios of a line joining two points and alsodiscuss about the equations of lines and planes in spaceunder different conditions, angle between two lines, twoplanes, a line and a plane, shortest distance between twoskew lines and distance of a point from a plane. Most ofthe above results are obtained in vector form. Nevertheless, we shall also translatethese results in the Cartesian form which, at times, presents a more clear geometricand analytic picture of the Direction Cosines and Direction Ratios of a LineFrom Chapter 10, recall that if a directed line L passing through the origin makesangles , and with x, y and z-axes, respectively, called direction angles, then cosineof these angles, namely, cos , cos and cos are called direction cosines of thedirected line we reverse the direction of L, then the direction angles are replaced by their supplements.

THREE DIMENSIONAL GEOMETRY 465 Hence, from (1), the d.c.’s of the line are 2 22 2 22 2 22, , a b c l m n abc abc abc =± =± =± ++ ++ ++ where, depending on the desired sign of k, either a positive or a negative sign is to be taken for l, m and n. For any line, if a, b, c are direction ratios of a line, then ka, kb, kc; k ≠ 0 is also a set of direction ratios.

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