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Vector Algebra

Question
CBSEENMA12034214
Wired Faculty App

Determine the area of the parallelogram, whose adjacent sides are given by the vectors 2 straight i with hat on top plus straight j with hat on top plus straight k with hat on top space and space straight i with hat on top space minus 2 straight j with hat on top plus space straight k with hat on top.

Solution

Let straight a with rightwards arrow on top space equals space 2 space straight i with hat on top space plus space straight j with hat on top space plus space straight k with hat on top comma space space space space space space straight b with rightwards arrow on top space equals space straight i with hat on top space minus space 2 space straight j with hat on top space plus 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 straight j with hat on top end cell cell straight k with hat on top end cell row 2 1 1 row 1 cell negative 2 end cell 1 end table close vertical bar space equals space straight i with hat on top space open vertical bar table row 1 1 row cell negative 2 end cell 1 end table close vertical bar minus straight j with hat on top space open vertical bar table row 2 1 row 1 1 end table close vertical bar plus straight k with hat on top open vertical bar table row 2 1 row 1 cell negative 2 end cell end table close vertical bar
             equals space left parenthesis 1 plus 2 right parenthesis space straight i with hat on top space minus space left parenthesis 2 minus 1 right parenthesis space straight j with hat on top space plus space left parenthesis negative 4 minus 1 right parenthesis space straight k with hat on top space equals space equals 3 straight i with hat on top minus straight j with hat on top space minus space 5 straight k with hat on top
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 left parenthesis 3 right parenthesis squared plus left parenthesis negative 1 right parenthesis squared plus left parenthesis negative 5 right parenthesis squared end root space equals space square root of 9 plus 1 plus 25 end root space equals space square root of 35
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 35 space space sq. space units.

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