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Sequences And Series

Question
CBSEENMA11014977
Wired Faculty App

Show that the points:
P (-1,-6, 10), Q (1, -3, 4), R (-5, -1, 1) and S (-7, -4, 7) are the vertices of a rhombus.

Solution
<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>
Using distance formula, we have
                         <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>
                                   equals space square root of left parenthesis negative 1 minus 1 right parenthesis squared plus left parenthesis negative 6 plus 3 right parenthesis squared plus left parenthesis 10 minus 4 right parenthesis squared end root space equals space square root of 4 plus 9 plus 36 end root space equals space square root of 49 space equals space 7            ...(i)
                             QR space equals space square root of left parenthesis straight x subscript 2 minus straight x subscript 3 right parenthesis squared plus left parenthesis straight y subscript 2 minus straight y subscript 3 right parenthesis squared plus left parenthesis straight z subscript 2 minus straight z subscript 3 right parenthesis squared end root
                            space space space space space space space space equals space square root of left parenthesis 1 plus 5 right parenthesis squared plus left parenthesis negative 3 plus 1 right parenthesis squared plus left parenthesis 4 minus 1 right parenthesis squared end root space equals space square root of 36 plus 4 plus 9 end root space equals square root of 49 space equals space 7                   ...(ii)
                           <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>
                               space space space space equals space square root of left parenthesis negative 5 plus 7 right parenthesis squared plus left parenthesis negative 1 plus 4 right parenthesis squared plus left parenthesis 1 minus 7 right parenthesis squared end root space equals space square root of 4 plus 9 plus 36 end root space equals space square root of 49 space equals space 7              ...(iii)
                          space space space SP space equals space square root of left parenthesis straight x subscript 4 minus straight x subscript 1 right parenthesis squared plus left parenthesis straight y subscript 4 minus straight y subscript 1 right parenthesis squared plus left parenthesis straight z subscript 4 minus straight z subscript 1 right parenthesis squared end root
                          space space space space space space space space equals space square root of left parenthesis negative 7 plus 1 right parenthesis squared plus left parenthesis negative 4 plus 6 right parenthesis squared plus left parenthesis 7 minus 10 right parenthesis squared end root space equals space square root of 36 plus 4 plus 9 end root space equals space square root of 49 space equals space 7              ...(iv) 
From (i), (ii), (iii) and (iv), PQ = QR = RS = SP
Hence, PQRS is a rhombus.

Some More Questions From Sequences and Series Chapter

The co-ordinates of a point A are (2, -3, 6). Write down the co-ordinates of seven points whose absolute values are the same as those of the co-ordinates of the given point.

Find the octant in which the following points lie:
A (-3,1, 2), B (-3, 1, -2), C (-3, -1,-2), D (3, -1, -2) and E (3,1, 2).

A point lies on the x-axis at a distance of 5 units towards OX'. What are its co-ordinates?

A point lies on z-axis at a distance of 3 units in the direction OZ. Write its co-ordinates.

A point lies on y-axis and is at a distance of 4 units in the direction OY'. Write its co-ordinates.

A point lies on XY-plane with distances 4 units along OX and 3 units along OY. Write its co-ordinates.

A point lies on YZ-plane with distances 4 units along OY and 5 units along OZ'. Write its co-ordinates.

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