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Vector Algebra

Question
CBSEENMA12033969

In the figure, M is the mid-point of AB and N is the mid-point of CD and O is the mid-point of MN. Prove that
(i) OA with rightwards arrow on top space plus space OB with rightwards arrow on top space plus space OC with rightwards arrow on top space plus space OD with rightwards arrow on top space equals space straight O with rightwards arrow on top
(ii) BC with rightwards arrow on top space plus space AD with rightwards arrow on top space equals space 2 space MN with rightwards arrow on top

Solution

We have
                          OA with rightwards arrow on top space plus space AM with rightwards arrow on top space equals space OM with rightwards arrow on top                             ...(1)
and                   OB with rightwards arrow on top space plus space BM with rightwards arrow on top space equals space OM with bar on top                               ...(2)
Adding (1) and (2) we get,
           OA with rightwards arrow on top space plus space OB with rightwards arrow on top space plus space AM with rightwards arrow on top space plus space BM with rightwards arrow on top space equals space 2 space OM with rightwards arrow on top
OA with rightwards arrow on top space plus OB with rightwards arrow on top space plus space AM with rightwards arrow on top space minus space AM with rightwards arrow on top space equals space 2 space OM with rightwards arrow on top space space space left square bracket because space space AM with rightwards arrow on top space equals space minus BM with rightwards arrow on top right square bracket
therefore space space space space space space space space OA with rightwards arrow on top space plus space OB with rightwards arrow on top space equals space 2 space OM with rightwards arrow on top 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... left parenthesis 3 right parenthesis
Similarly space space space space OC with rightwards arrow on top space plus space OD with rightwards arrow on top space equals space 2 space ON with rightwards arrow on top 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... left parenthesis 4 right parenthesis space space space space space space space space space space space space space space space space space space space space space space
Adding (3) and (4), we get,
               OA with rightwards arrow on top plus OB with rightwards arrow on top plus OC with rightwards arrow on top plus OD with rightwards arrow on top space equals space 2 space open parentheses OM with rightwards arrow on top plus ON with rightwards arrow on top close parentheses
                                                  equals space 2 space left parenthesis OM with rightwards arrow on top space minus space OM with rightwards arrow on top right parenthesis            open square brackets because space space space OM with rightwards arrow on top space equals space minus ON with rightwards arrow on top close square brackets
                                                   equals space 2 space left parenthesis 0 with rightwards arrow on top right parenthesis
         therefore space space space space OA with rightwards arrow on top space plus space OB with rightwards arrow on top space plus OC with rightwards arrow on top space plus space OD with rightwards arrow on top space equals space 0 with rightwards arrow on top
Also,     BC with rightwards arrow on top space equals space BM with rightwards arrow on top space plus space MN with rightwards arrow on top space plus space NC with rightwards arrow on top space
and       AD with rightwards arrow on top space equals space AM with rightwards arrow on top space plus space MN with rightwards arrow on top space plus space ND with rightwards arrow on top
Adding,                    BC with rightwards arrow on top space plus space AD with rightwards arrow on top space equals space left parenthesis BM with rightwards arrow on top space plus space AM with rightwards arrow on top right parenthesis space plus space 2 space MN with rightwards arrow on top space plus space left parenthesis NC with rightwards arrow on top space plus space ND with rightwards arrow on top right parenthesis
                                                  equals space open parentheses negative AM with rightwards arrow on top space plus space AM with rightwards arrow on top close parentheses space plus space 2 space MN with rightwards arrow on top space plus left parenthesis space stack NC space with rightwards arrow on top space minus space NC with rightwards arrow on top right parenthesis
              therefore space space space space space space space space BC with rightwards arrow on top space plus space AD with rightwards arrow on top space equals space 0 with rightwards arrow on top space plus space 2 space MN with rightwards arrow on top space plus space 0 with rightwards arrow on top
                                     equals 2 space MN with rightwards arrow on top