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Vector Algebra

Question
CBSEENMA12034255

Show that left parenthesis straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top right parenthesis space cross times space left parenthesis straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top right parenthesis space equals space 2 space straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top .
OR
Show that left parenthesis straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top right parenthesis space cross times space left parenthesis straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top right parenthesis space equals space 2 space straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top and give a geometrical interpretation of it.

Solution

L.H.S. = left parenthesis straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top right parenthesis space cross times space left parenthesis straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top right parenthesis space equals space straight a with rightwards arrow on top space cross times space straight a with rightwards arrow on top space plus space straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space minus space straight b with rightwards arrow on top space cross times space straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top space cross times space straight b with rightwards arrow on top
            equals space straight a with rightwards arrow on top space cross times space straight a with rightwards arrow on top space plus space straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space plus space straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space minus space straight b with rightwards arrow on top space equals space 0 space plus space 2 straight a with rightwards arrow on top cross times straight b with rightwards arrow on top space minus space 0 with rightwards arrow on top space equals space 2 space straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top
equals space straight R. straight H. straight S.
Gemetrical Interpretation
Let O be the point of intersection of diagonals AC and BD of || gm ABCD.
           Let  AO with rightwards arrow on top space equals straight a with rightwards arrow on top comma space space OD with rightwards arrow on top space equals space straight b with rightwards arrow on top
therefore space space space space space OB with rightwards arrow on top space equals space minus straight b with rightwards arrow on top
therefore space space space AB with rightwards arrow on top space equals space AO with rightwards arrow on top space plus space OB with rightwards arrow on top space equals space straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top
       AD with rightwards arrow on top space equals AO with rightwards arrow on top space plus space OD with rightwards arrow on top space equals straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top

therefore space space space left parenthesis straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top right parenthesis space cross times space left parenthesis straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top right parenthesis space equals space vector space area space of space vertical line vertical line space gm space ABCD
Also comma space straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space equals space vector space area space of space vertical line vertical line space gm space whose space sides space area space straight a with rightwards arrow on top space space and space straight b with rightwards arrow on top.
therefore space space space space left parenthesis straight a with rightwards arrow on top minus straight b with rightwards arrow on top right parenthesis space cross times space left parenthesis straight a with rightwards arrow on top space plus straight b with rightwards arrow on top right parenthesis space equals space 2 space straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top implies that the area of || gm ABCD is twice the area of the || gm whose adjacent sides are semi-diagonals of the first parallelogram.