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Vector Algebra

Question
CBSEENMA12034216
Wired Faculty App

Find the area of the parallelogram whose diagonals are
straight a with rightwards arrow on top space equals space 3 space straight i with hat on top space plus space straight j with hat on top space minus space 2 space straight k with hat on top space space space and space straight b with rightwards arrow on top space equals space straight i with hat on top space minus space 3 space straight j with hat on top space plus space 4 space straight k with hat on top.

Solution
Let ABCD be the parallelogram such that
straight a with rightwards arrow on top space equals space AC with rightwards arrow on top space equals space 3 straight i with hat on top space plus space straight j with hat on top space minus space 2 space straight k with hat on top comma space space space space straight b with rightwards arrow on top space equals space BD with rightwards arrow on top space equals space straight i with hat on top space minus space 3 space straight j with hat on top space plus space 4 space straight k with hat on top
            Vectors of || gm ABCD = Vector area of ∆ ABC + Vector area of ∆ ACD

                equals space 1 half AB with rightwards arrow on top space cross times space AC with rightwards arrow on top space plus space 1 half AC with rightwards arrow on top space cross times space AD with rightwards arrow on top equals space 1 half left parenthesis negative AC with rightwards arrow on top space cross times space AB with rightwards arrow on top space plus space AC with rightwards arrow on top space cross times space AD with rightwards arrow on top right parenthesis equals space 1 half AC with rightwards arrow on top space cross times space left parenthesis negative AB with rightwards arrow on top space plus space AD with rightwards arrow on top right parenthesis equals space 1 half AC with rightwards arrow on top space cross times space left parenthesis BA with rightwards arrow on top space plus space AD with rightwards arrow on top right parenthesis space equals space 1 half AC with rightwards arrow on top space cross times space BD with rightwards arrow on top space equals space 1 half straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top equals space 1 half left parenthesis 3 straight i with hat on top space plus space space straight j with hat on top minus space 2 space straight k with hat on top right parenthesis space cross times left parenthesis straight i with hat on top space minus space 3 space straight j with hat on top space plus space 4 space straight k with hat on top right parenthesis space equals space 1 half open vertical bar table row cell straight i with hat on top end cell cell straight j with hat on top end cell cell straight k with hat on top end cell row 3 1 cell negative 2 end cell row 1 cell negative 3 end cell cell space 4 end cell end table close vertical bar space equals space 1 half left square bracket left parenthesis 4 minus 6 right parenthesis space straight i with hat on top space minus space left parenthesis 12 plus 2 right parenthesis space straight j with hat on top plus space left parenthesis negative 5 minus 1 right parenthesis straight k with hat on top equals space 1 half left parenthesis negative 2 space straight i with hat on top space plus space 14 space straight j with hat on top space minus space 10 space straight k with hat on top right parenthesis space equals space minus straight i with hat on top space minus space 7 space straight j with hat on top space minus space 5 space straight k with hat on top therefore space space space area space of space vertical line vertical line space gm thin space ABCD space equals space open vertical bar negative straight i with hat on top minus 7 space straight j with hat on top space minus space 5 space straight k with hat on top close vertical bar space equals square root of 1 plus 49 plus 25 end root space equals square root of 75 space equals space 5 space square root of 3 space sq. space units. space

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