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Vector Algebra

Question
CBSEENMA12034156
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Using vectors, prove that a parallelogram whose diagonals are equal is a rectangle. 

Solution
Let OACB be a parallelogram such that OC = AB

Let OA with rightwards arrow on top space equals space straight a with rightwards arrow on top comma space space OB with rightwards arrow on top space equals space straight b with rightwards arrow on top
Now,  OC = AB
rightwards double arrow space space space OC squared space equals space AB squared rightwards double arrow space space open parentheses OA with rightwards arrow on top plus AC with rightwards arrow on top close parentheses squared space equals open parentheses AO with rightwards arrow on top space plus OB with rightwards arrow on top close parentheses squared rightwards double arrow space space space open parentheses OA with rightwards arrow on top space plus space OB with rightwards arrow on top close parentheses squared space equals space open parentheses negative OA with rightwards arrow on top space plus space OB with rightwards arrow on top close parentheses squared rightwards double arrow space space left parenthesis straight a with rightwards arrow on top space plus straight b with rightwards arrow on top right parenthesis squared space equals space left parenthesis negative straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top right parenthesis squared rightwards double arrow space space straight a with rightwards arrow on top squared 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 minus space 2 space straight a with rightwards arrow on top. space straight b with rightwards arrow on top rightwards double arrow space space 2 space straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space minus 2 space straight a with rightwards arrow on top. space straight b with rightwards arrow on top space space space rightwards double arrow space space space 4 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 straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space 0 with rightwards arrow on top rightwards double arrow space space space straight a with rightwards arrow on top space and space straight b with rightwards arrow on top space are space perpendicular. space space space space space space rightwards double arrow space space space angle AOB space equals space 90 degree rightwards double arrow space space space OACB space is space straight a space rectangle.
           

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