Sequences And Series

Question
CBSEENMA11015063

If P(x1, y1, z1) and Q(x,y,z2) be the end points of a line segment PQ and let R be a point on PQ dividing PQ in the ratio m : n internally, then prove that the co-ordinates of R are:
              space space open parentheses fraction numerator mx subscript 2 plus nx subscript 1 over denominator straight m plus straight n end fraction comma space fraction numerator my subscript 2 plus ny subscript 1 over denominator straight m plus straight n end fraction comma space fraction numerator mz subscript 2 plus nz subscript 1 over denominator straight m plus straight n end fraction close parentheses

Solution
PROOF: Let R(x, y, z) be the points which divides the line segment PQ in the ratio m : n internally. Draw PL, QN and RM perpendiculars to XY-plane as shown in figure. Through R, draw a line SRT parallel to the line LMN, produce LP to S and T lies on the line QN.
     
Clearly, SLMR and RMNT are parallelograms,
Also space space straight capital delta space PSR space and space straight capital delta space QTR are similar.
rightwards double arrow           straight m over straight n equals PR over QR equals SP over QT equals fraction numerator SL minus PL over denominator QN minus TN end fraction
                      equals space fraction numerator RM minus PL over denominator QN minus RM end fraction equals fraction numerator straight z minus straight z subscript 1 over denominator straight z subscript 2 minus straight z end fraction
or                straight z equals fraction numerator mz subscript 2 plus nz subscript 1 over denominator straight m plus straight n end fraction
Similarly, we have straight x equals fraction numerator mx subscript 2 plus nx subscript 1 over denominator straight m plus straight n end fraction space space and space straight y equals fraction numerator my subscript 2 plus ny subscript 1 over denominator straight m plus straight n end fraction
Hence, R is <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>
Similarly, for external division in the ratio m : n, the co-ordinate of R:
                      open parentheses fraction numerator mx subscript 2 minus nx subscript 1 over denominator straight m minus straight n end fraction comma space fraction numerator my subscript 2 minus ny subscript 1 over denominator straight m minus straight n end fraction comma space fraction numerator mz subscript 2 minus nz subscript 1 over denominator straight m minus straight n end fraction close parentheses