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Vector Algebra

Question
CBSEENMA12033948
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ABCD is a parallelogram. If AB with rightwards arrow on top space equals space straight a with rightwards arrow on top comma space space space BC with rightwards arrow on top space equals space straight b with rightwards arrow on top comma space   then
(i)    Show that AC with rightwards arrow on top space equals straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top comma space space space space BD with rightwards arrow on top space equals space straight b with rightwards arrow on top space minus space straight a with rightwards arrow on top.
(ii) What is the geometrical significance of the relation
                      open vertical bar straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top close vertical bar space equals space open vertical bar straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top close vertical bar ?

Solution

Here AB with rightwards arrow on top space equals space stack straight a comma with rightwards arrow on top space space BC with rightwards arrow on top space equals space straight b with rightwards arrow on top
(i) In increment ABC comma space space 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 straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top
In increment ABD comma space space AB with rightwards arrow on top space plus space BD with rightwards arrow on top space equals space AD with rightwards arrow on top
   therefore space space space space BD with rightwards arrow on top space equals space AD with rightwards arrow on top space minus space AB with rightwards arrow on top space equals space BC with rightwards arrow on top space minus space AB with rightwards arrow on top                                                  open square brackets because space space space AD with rightwards arrow on top space equals space BC with rightwards arrow on top close square brackets

                         equals space straight b with rightwards arrow on top space minus space straight a with rightwards arrow on top
(ii)
       open vertical bar straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top close vertical bar space equals space open vertical bar straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top close vertical bar
rightwards double arrow space space space space space open vertical bar straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top close vertical bar space equals space open vertical bar straight b with rightwards arrow on top space minus space straight a with rightwards arrow on top close vertical bar space as space open vertical bar negative straight a with rightwards arrow on top close vertical bar space equals space open vertical bar straight a with rightwards arrow on top close vertical bar therefore space space space space space space space space open vertical bar AC with rightwards arrow on top close vertical bar space equals space open vertical bar BD with rightwards arrow on top close vertical bar rightwards double arrow space space space space space space space space space space AC space equals space BD space space space space space space space space space space space space space space space space space space rightwards double arrow space space space space space space diagonals space of space vertical line vertical line space gm space ABCD space are space equal therefore space space space space space geometrically space open vertical bar straight a plus straight b close vertical bar space equals open vertical bar straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top close vertical bar space means space ABCD space is space straight a space rectangle. space

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