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Vector Algebra

Question
CBSEENMA12034096
Wired Faculty App

For any two vectors straight a with rightwards arrow on top space and space straight b with rightwards arrow on top, we always have open vertical bar straight a with rightwards arrow on top plus straight b with rightwards arrow on top close vertical bar space less or equal than space open vertical bar straight a with rightwards arrow on top close vertical bar space plus space open vertical bar straight b with rightwards arrow on top close vertical bar

Solution

Assume that open vertical bar straight a with rightwards arrow on top close vertical bar space not equal to space 0 comma space space space open vertical bar straight b with rightwards arrow on top close vertical bar space not equal to space 0 space space space space space space space left square bracket because space space result space is space true space for space straight a with rightwards arrow on top space equals space 0 space or space straight b with rightwards arrow on top space equals space 0 right square bracket
             open vertical bar straight a with rightwards arrow on top plus straight b with rightwards arrow on top close vertical bar squared space equals space left parenthesis straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top right parenthesis squared space equals space straight a with rightwards arrow on top squared space plus space straight b with rightwards arrow on top squared space plus space 2 space straight a with rightwards arrow on top space straight b with rightwards arrow on top
                           equals space open vertical bar straight a with rightwards arrow on top close vertical bar squared space plus space open vertical bar straight b with rightwards arrow on top close vertical bar squared space plus space 2 space open vertical bar straight a with rightwards arrow on top close vertical bar space open vertical bar straight b with rightwards arrow on top close vertical bar space cos space straight theta less or equal than space 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 space plus space 2 space open vertical bar straight a with rightwards arrow on top close vertical bar space. space open vertical bar straight b 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 left square bracket because space cos space straight theta space less or equal than space 1 right square bracket space equals space left parenthesis open vertical bar straight a with rightwards arrow on top close vertical bar space plus space open vertical bar straight b with rightwards arrow on top close vertical bar right parenthesis squared
therefore space space space space open vertical bar straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top close vertical bar squared space space less or equal than space open parentheses open vertical bar straight a with rightwards arrow on top close vertical bar space plus space open vertical bar straight b with rightwards arrow on top close vertical bar close parentheses squared space space space space space space space space space open vertical bar straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top close vertical bar space less or equal than space open vertical bar straight a with rightwards arrow on top close vertical bar space plus space open vertical bar straight b with rightwards arrow on top close vertical bar

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