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Vector Algebra

Question
CBSEENMA12034212
Wired Faculty App

Determine the area of the parallelogram, whose adjacent sides are given by the vectors
straight i with hat on top minus straight j with hat on top minus straight k with hat on top space and space space 3 space straight i with hat on top space plus space 4 space straight j with hat on top space minus space 5 space straight k with hat on top.

Solution

Let     straight a with rightwards arrow on top space equals space straight i with hat on top space minus space straight j with hat on top space minus space straight k with hat on top comma space space space straight b with rightwards arrow on top space equals space 3 space straight i with hat on top space plus space 4 space straight j with hat on top space minus space 5 space straight k with hat on top
straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space equals space open vertical bar table row cell straight i with hat on top end cell cell space space space straight j with hat on top end cell cell straight k with hat on top end cell row 1 cell negative 1 end cell cell negative 1 end cell row 3 cell space space 4 end cell cell negative 5 end cell end table close vertical bar
                         equals straight i with hat on top space open vertical bar table row cell negative 1 end cell cell negative 1 end cell row 4 cell negative 5 end cell end table close vertical bar space minus space straight j with hat on top space open vertical bar table row 1 cell negative 1 end cell row 3 cell negative 5 end cell end table close vertical bar space plus straight k with hat on top space open vertical bar table row 1 cell negative 1 end cell row 3 4 end table close vertical bar equals space left parenthesis 5 plus 4 right parenthesis space straight i with hat on top space minus space left parenthesis negative 5 plus 3 right parenthesis space straight j with hat on top space plus space left parenthesis 4 plus 3 right parenthesis space straight k with hat on top space equals space 9 space straight i with hat on top space plus space 2 space straight j with hat on top space plus space 7 space straight k with hat on top
open vertical bar straight a with rightwards arrow on top space cross times straight b with rightwards arrow on top close vertical bar space equals space square root of left parenthesis 9 right parenthesis squared plus left parenthesis 2 right parenthesis squared plus left parenthesis 7 right parenthesis squared end root space equals space square root of 81 plus 4 plus 49 end root space equals space square root of 134
therefore space space space area space of space parallelogram space equals space open vertical bar straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top close vertical bar space equals space square root of 134 space sq. space units.

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