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Vector Algebra

Question
CBSEENMA12033971

ABCD is a parallelogram. If L and M are the mid-point of BC and DC respectively, then express AL with rightwards arrow on top space and space AM with rightwards arrow on top in terms of AB with rightwards arrow on top space and space AD with rightwards arrow on top.

Solution

Take A as the origin. Let straight b with rightwards arrow on top space comma space straight d with rightwards arrow on top be position vectors of B and D respectively such that
    AB with rightwards arrow on top space equals space straight b with rightwards arrow on top space AD with rightwards arrow on top space equals space straight d with rightwards arrow on top
Now,    AL with rightwards arrow on top space equals space AB with rightwards arrow on top space plus space BL with rightwards arrow on top
                  equals space AB with rightwards arrow on top space plus space 1 half BC with rightwards arrow on top
equals space AB with rightwards arrow on top space plus space 1 half AD with rightwards arrow on top space equals space straight b plus fraction numerator straight d with rightwards arrow on top over denominator 2 end fraction
therefore space space space space space space space position space vector space of space straight L space is space straight b with rightwards arrow on top space plus space fraction numerator straight d with rightwards arrow on top over denominator 2 end fraction
Again AM with rightwards arrow on top space equals space AD with rightwards arrow on top space plus space DM with rightwards arrow on top space equals space AD with rightwards arrow on top space plus space 1 half space DC with rightwards arrow on top space equals space AD with rightwards arrow on top space plus space 1 half AB with rightwards arrow on top
                                 equals space straight d with rightwards arrow on top space plus space 1 half space straight b with rightwards arrow on top

AC with rightwards arrow on top space equals space AB with rightwards arrow on top space plus space BC with rightwards arrow on top space equals space AB with rightwards arrow on top space plus space AD with rightwards arrow on top space equals space straight b with rightwards arrow on top space plus space straight d with rightwards arrow on top
                                 ...(1)
AL with rightwards arrow on top space plus space AM with rightwards arrow on top space equals space open parentheses straight b with rightwards arrow on top space plus space fraction numerator straight d with rightwards arrow on top over denominator 2 end fraction close parentheses space plus space open parentheses straight d with rightwards arrow on top space plus space fraction numerator straight b with rightwards arrow on top over denominator 2 end fraction close parentheses space equals space 3 over 2 straight b with rightwards arrow on top space plus space 3 over 2 straight d with rightwards arrow on top
space space space space space space space space space space space space space space space space space space equals space 3 over 2 left parenthesis straight b with rightwards arrow on top space plus space straight d with rightwards arrow on top right parenthesis space equals space 3 over 2 AC with rightwards arrow on top space space space space space space space space space space space space space space space space space space left square bracket because space of space left parenthesis 1 right parenthesis right square bracket