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Vector Algebra

Question
CBSEENMA12034088
Wired Faculty App

Prove that two vectors straight a with rightwards arrow on top space and space straight b with rightwards arrow on top are perpendicular if 
       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 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

Solution
because space space space space space 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 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 left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top right parenthesis squared space 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 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 open square brackets because space open vertical bar straight a with rightwards arrow on top close vertical bar squared space equals space straight a with rightwards arrow on top squared close square brackets therefore space space space 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 space equals space straight a with rightwards arrow on top squared space plus space straight b with rightwards arrow on top squared space space space space rightwards double arrow space 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 rightwards double arrow space space space straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space 0 therefore space space space space straight a with rightwards arrow on top space and space straight b with rightwards arrow on top space are space perpendicular.

Tips: -

By writing the steps in the reverse order, we can prove that 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 open vertical bar straight a with rightwards arrow on top close vertical bar squared plus open vertical bar straight b with rightwards arrow on top close vertical bar squared only if  straight a with rightwards arrow on top space and space straight b with rightwards arrow on top space are space perpendicular.

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