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Vector Algebra

Question
CBSEENMA12034086
Wired Faculty App

Prove that
left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top right parenthesis. space open parentheses straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top close parentheses space equals 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
if and only if straight a with rightwards arrow on top comma space straight b with rightwards arrow on top are orthogonal.

Solution
(i) Assume that
           left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top right parenthesis. space left parenthesis straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top right parenthesis space equals 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 
therefore space space space space space space straight a with rightwards arrow on top. space straight a with rightwards arrow on top space plus space straight a with rightwards arrow on top. space straight b with rightwards arrow on top. space straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top space. space straight b with rightwards arrow on top space equals 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 rightwards double arrow space space space space space open vertical bar straight a with rightwards arrow on top close vertical bar squared space plus space straight a with rightwards arrow on top. space straight b with rightwards arrow on top space plus space straight a with rightwards arrow on top. space straight b with rightwards arrow on top space plus space open vertical bar straight b with rightwards arrow on top close vertical bar squared space equals 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 rightwards double arrow space space space 2 space straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space 0 space space space space space space space rightwards double arrow space space space space space straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space 0 rightwards double arrow space space space space straight a with rightwards arrow on top. space straight b with rightwards arrow on top space are space orthogonal. space
(ii) Assume that straight a with rightwards arrow on top comma space straight b with rightwards arrow on top  are orthogonal
   therefore space space space space space straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space 0                                                                  ...(1)
Now left parenthesis straight a with rightwards arrow on top plus straight b 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 right parenthesis space equals space straight a with rightwards arrow on top. space straight a with rightwards arrow on top space plus space straight a with rightwards arrow on top. space straight b with rightwards arrow on top space plus space straight b 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
                            equals space open vertical bar straight a with rightwards arrow on top close vertical bar squared plus straight a with rightwards arrow on top. space straight b with rightwards arrow on top space plus space straight a with rightwards arrow on top. space straight b with rightwards arrow on top space plus space open vertical bar straight b with rightwards arrow on top close vertical bar squared space equals space open vertical bar straight a with rightwards arrow on top close vertical bar squared plus 0 plus 0 plus open vertical bar straight b with rightwards arrow on top close vertical bar squared space space space space space space left square bracket because space of space left parenthesis 1 right parenthesis right square bracket
Now,    left parenthesis straight a with rightwards arrow on top plus straight b 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 right parenthesis space equals 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
Hence the result. 

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