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File: Geometry Pdf 166381 | Pseb Class 12 Math Textbook Englishpart 2 Chapter 11
three dimensional geometry 463 chapter 11 three dimensional geometry vthe moving power of mathematical invention is not reasoning but imagination a demorgan v 11 1 introduction in class xi while ...

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                                                                            THREE DIMENSIONAL GEOMETRY         463
                                                                                             Chapter 11
                                THREE DIMENSIONAL GEOMETRY
                                           vThe 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 a, b and g with x, y and z-axes, respectively, called direction angles, then cosine
                             of these angles, namely, cos a, cos b and cos g 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.,  p - a ,  p - b and  p - g . 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
                            464     MATHEMATICS
                                                              Fig 11.1
                                Note that a given line in space can be extended in two opposite directions and so it
                            has two sets of direction cosines. In order to have a unique set of direction cosines for
                            a given line in space, we must take the given line as a directed line. These unique
                            direction cosines are denoted by l, m and n.
                            Remark If the given line in space does not pass through the origin, then, in order to find
                            its direction cosines, we draw a line through the origin and parallel to the given line.
                            Now take one of the directed lines from the origin and find its direction cosines as two
                            parallel line have same set of direction cosines.
                                Any three numbers which are proportional to the direction cosines of a line are
                            called the direction ratios of the line. If l, m, n are direction cosines and a, b, c are
                            direction ratios of a line, then a  ll, blm and c  ln, for any nonero l	Î	R.
                             ANote  Some authors also call direction ratios as direction numbers.
                                Let a, b, c be direction ratios of a line and let l, m and n be the direction cosines
                            d.cs of the line. Then
                                                       l    m   n = k  say, k being a constant.
                                                       a    b     c
                            Therefore                 l  ak, m bk, n  ck                           ... 1
                                                    2    2    2
                            But                    l   m   n  1
                                                2  2    2    2
                            Therefore          k  a   b   c  1
                            or                               k  ±       1
                                                                      a2 +b2 +c2
                                                                     THREE DIMENSIONAL GEOMETRY      465
                           Hence, from 1, the d.c.s of the line are
                                       l =±     a      ,m=±        b      ,n=±       c
                                             a2 +b2+c2         a2 +b2+c2         a2 +b2+c2
                           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 ¹ 0 is also a
                           set of direction ratios. So, any two sets of direction ratios of a line are also proportional.
                           Also, for any line there are infinitely many sets of direction ratios.
                           11.2.1  Relation between the direction cosines of a line
                           Consider a line RS with direction cosines l, m, n. Through
                           the origin draw a line parallel to the given line and take a
                           point x, y, z on this line. From  draw a perpendicular
                           A on the x-axis Fig. 11.2.
                           Let O  r. Thencosa = OA = x . This gives x  lr..
                                                    O r
                           Similarly,                 y mr and z  nr
                                              2   2    2   2  2    2   2
                           Thus              x   y   z  r  l   m   n 
                                              2   2    2   2
                           But               x   y   z  r
                           Hence            l2 + m2 + n2 = 1                            Fig 11.2
                           11.2.2  Direction cosines of a line passing through two points
                           Since one and only one line passes through two given points, we can determine the
                           direction cosines of a line passing through the given points x , y , z  and x , y , z 
                           as follows Fig 11.3 a.                              1  1 1        2  2  2
                                                              Fig 11.3
                                        466       MATHEMATICS
                                             Let l, m, n be the direction cosines of the line  and let it makes angles a, b and  g
                                        with the x, y and z-axis, respectively.
                                             Draw perpendiculars from  and  to XY-plane to meet at R and S. Draw a
                                        perpendicular from  to S to meet at N. Now, in right angle triangle N, ÐN
                                        g		Fig 11.3 b.
                                                                              N z -z
                                        Therefore,                 cosg            = 2        1
                                                                                        
                                                                              x -x                       y -y
                                        Similarly                 cosa  2           1 and cosb= 2              1
                                                                                                         
                                             Hence, the direction cosines of the line segment joining the points x , y , z  and
                                        x , y , z  are                                                                                 1   1   1
                                             2   2    2
                                                                              x    - x       y   - y       z   - z
                                                                                2       1 ,   2       1 ,   2       1
                                                                                                          
                                        where                                              2                  2                 2
                                                                                x    - x  + y - y  + z - z
                                                                                   2       1            2      1          2      1
                                               Note  The direction ratios of the line segment joining x , y , z  and x , y , z 
                                         A                                                                               1   1   1            2   2   2
                                         may be taken as
                                                         x  x , y  y , z  z  or x  x , y  y , z  z
                                                          2     1    2     1   2     1      1     2    1     2   1     2
                                        Example 1If a line makes angle 0, 60 and 30 with the positive direction of x, y and
                                        z-axis respectively, find its direction cosines.
                                        Solution Let the d.c.s of the lines be l , m, n. Then l  cos 00  0, m  cos 600   1 ,
                                                                                                                                                      2
                                                      0       3
                                        n  cos 30    2 .
                                        Example 2If a line has direction ratios 2,  1,  2, determine its direction cosines.
                                        Solution Direction cosines are
                                                                2               ,              -1               ,              -2
                                                      2           2          2         2          2          2         2          2          2
                                                    2 +-1+-2                     2 + -1 + -2                  2 + -1 +-2
                                        or        2, -1, -2
                                                  3 3        3
                                        Example 3 Find the direction cosines of the line passing through  the two points
                                         2, 4,  5 and 1, 2, 3.
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