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Vector Algebra

Question
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In a triangle OAB , ∠AOB = 90°. If P and Q are the points of trisection of AB, prove that OP squared space plus space OQ squared space equals space 5 over 9 AB squared

Solution

Take O as origin,  OA with rightwards arrow on top space equals 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
because space space space angle AOB space equals space 90 degree space space space space space space space space space space space space because space space OA with rightwards arrow on top. space OB with rightwards arrow on top space equals space 0 space space space space space space space space 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
Let P divide AB in the ratio 1 : 2
because space straight P. straight V. space of space straight P space equals space fraction numerator 2 straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top over denominator 3 end fraction
Let Q divide AB in the ratio 2 : 1
because space straight P. straight V. space of space straight Q space equals space fraction numerator straight a with rightwards arrow on top space plus space 2 space straight b with rightwards arrow on top over denominator 3 end fraction
L.H.S. = OP squared plus OQ squared space equals space open parentheses fraction numerator 2 space straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top over denominator 3 end fraction close parentheses squared space plus space open parentheses fraction numerator straight a with rightwards arrow on top space plus 2 space straight b with rightwards arrow on top over denominator 3 end fraction close parentheses space equals space 1 over 9 open square brackets left parenthesis 2 space straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top right parenthesis space squared space plus space left parenthesis straight a with rightwards arrow on top space plus space 2 space straight b with rightwards arrow on top right parenthesis squared close square brackets
            equals space 1 over 9 open square brackets 4 straight a with rightwards arrow on top squared space plus space straight b with rightwards arrow on top squared space plus space 4 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 squared space plus space 4 space straight b with rightwards arrow on top squared space minus space 4 space straight a with rightwards arrow on top. space straight b with rightwards arrow on top close square brackets equals space 1 over 9 left parenthesis 5 straight a with rightwards arrow on top squared space plus space 5 straight b with rightwards arrow on top squared right parenthesis 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 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 equals 5 over 9 left parenthesis straight a with rightwards arrow on top squared plus space straight b with rightwards arrow on top squared right parenthesis

R.H.S. = 5 over 9 AB squared space equals space 5 over 9 AB with rightwards arrow on top squared space equals space 5 over 9 left parenthesis straight P. straight V. space of space straight B space minus space straight P. straight V. space of space straight A right parenthesis squared
           equals space 5 over 9 left parenthesis straight b with rightwards arrow on top space minus space straight a with rightwards arrow on top right parenthesis squared space equals space 5 over 9 left square bracket straight b with rightwards arrow on top squared space plus space straight a with rightwards arrow on top squared space minus space space 2 space straight b with rightwards arrow on top. space straight a with rightwards arrow on top right square bracket space equals space 5 over 2 left parenthesis straight a with rightwards arrow on top squared space plus space straight b with rightwards arrow on top squared right parenthesis space space space space left square bracket because space of space left parenthesis 1 right parenthesis right square bracket because space space space space straight L. straight H. straight S. space equals space straight R. straight H. straight S.

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