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Vector Algebra

Question
CBSEENMA12034133
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If straight a with rightwards arrow on top comma space straight b with rightwards arrow on top space and space straight c with rightwards arrow on top are three mutually perpendicular vectors of equal magnitude find the angle between straight a with rightwards arrow on top space and space left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top plus straight c with rightwards arrow on top right parenthesis.

Solution

We have
open vertical bar straight a with rightwards arrow on top close vertical bar space equals open vertical bar straight b with rightwards arrow on top close vertical bar space equals space open vertical bar straight c with rightwards arrow on top close vertical bar space and space straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space straight b with rightwards arrow on top. space straight c with rightwards arrow on top space equals space straight c with rightwards arrow on top. space straight a with rightwards arrow on top space equals 0                     ...(1)
Now open vertical bar straight a with rightwards arrow on top plus straight b with rightwards arrow on top plus straight c with rightwards arrow on top close vertical bar squared space space equals space left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top plus straight c with rightwards arrow on top right parenthesis squared space equals left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top plus straight c with rightwards arrow on top right parenthesis. space left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top plus straight c with rightwards arrow on top right parenthesis.
                    equals space straight a with rightwards arrow on top. space straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top. space straight b with rightwards arrow on top space plus space straight c with rightwards arrow on top. space straight c with rightwards arrow on top space equals open vertical bar straight a with rightwards arrow on top close vertical bar squared plus space open vertical bar straight b with rightwards arrow on top close vertical bar squared plus space open vertical bar straight c with rightwards arrow on top close vertical bar squared space space space space space space space space open square brackets because space of space left parenthesis 1 right parenthesis close square brackets equals space open vertical bar straight a with rightwards arrow on top close vertical bar squared plus space open vertical bar straight a with rightwards arrow on top close vertical bar squared plus space open vertical bar straight a with rightwards arrow on top close vertical bar squared space equals 3 space open vertical bar straight a with rightwards arrow on top close vertical bar squared space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because space of space left parenthesis 1 right parenthesis close square brackets space therefore space space space open vertical bar straight a with rightwards arrow on top plus straight b with rightwards arrow on top plus straight c with rightwards arrow on top close vertical bar space equals space square root of 3 space open vertical bar straight a with rightwards arrow on top close vertical bar space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 2 right parenthesis space
Let straight theta be angle between <pre>uncaught exception: <b>file_put_contents(/home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/web/../../../../../../formulas/5a/9e/e6cfd017e47b71817b27babff0d8.ini): failed to open stream: Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/sys/io/File.class.php at line #12file_put_contents(/home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/web/../../../../../../formulas/5a/9e/e6cfd017e47b71817b27babff0d8.ini): failed to open stream: Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/sys/io/File.class.php line 12<br />#0 [internal function]: _hx_error_handler(2, 'file_put_conten...', '/home/config_ad...', 12, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/sys/io/File.class.php(12): file_put_contents('/home/config_ad...', 'mml=<math xmlns...') #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(48): sys_io_File::saveContent('/home/config_ad...', 'mml=<math xmlns...') #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(112): com_wiris_util_sys_Store->write('mml=<math xmlns...') #4 /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...') #5 /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) #6 /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) #7 {main}</pre>
therefore space space space straight a with rightwards arrow on top. space left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top plus straight c with rightwards arrow on top right parenthesis space equals space open vertical bar straight a with rightwards arrow on top close vertical bar space open vertical bar straight a with rightwards arrow on top plus space straight b with rightwards arrow on top plus straight c with rightwards arrow on top close vertical bar space cos space straight theta therefore space space space space space space space space space space space space straight a with rightwards arrow on top. space straight a with rightwards arrow on top space equals space open vertical bar straight a with rightwards arrow on top close vertical bar space. space square root of 3 space open vertical bar straight a with rightwards arrow on top close vertical bar space space cos space straight theta space space space space space space space space space space space space space space space space space space space open square brackets because space of space left parenthesis 1 right parenthesis comma space left parenthesis 2 right parenthesis close square brackets rightwards double arrow space space space open vertical bar straight a with rightwards arrow on top close vertical bar squared space equals space square root of 3 space space open vertical bar straight a with rightwards arrow on top close vertical bar squared space cos space straight theta space space rightwards double arrow space space cos space straight theta space equals space fraction numerator 1 over denominator square root of 3 end fraction therefore space space space space straight theta space equals space cos to the power of negative 1 end exponent open parentheses fraction numerator 1 over denominator square root of 3 end fraction close parentheses. space space
                 

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