Question

Find the area of the parallelogram having two adjacent sides OA and OB where O is the origin and the position vectors of A and B are respectively
$straight a with rightwards arrow on top space equals space straight i with hat on top plus 2 space straight j with hat on top space plus space 3 space straight k with hat on top$ and $straight b with rightwards arrow on top space equals space minus 3 space straight i with hat on top space minus space 2 space straight j with hat on top space plus space straight k with hat on top$

Solution

straight a with rightwards arrow on top space equals space straight i with hat on top space plus space 2 space straight j with hat on top space plus space 3 space straight k with hat on top comma space space space straight b with rightwards arrow on top space equals space minus 3 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 1 2 3 row cell negative 3 end cell cell negative 2 end cell 1 end table close vertical bar space equals space left parenthesis 2 plus 6 right parenthesis space straight i with hat on top space minus space left parenthesis 1 plus 9 right parenthesis straight j with hat on top space plus space left parenthesis negative 2 plus 6 right parenthesis space straight k with hat on top space equals space 8 straight i with hat on top space minus space 10 straight j with hat on top space plus space 4 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 8 right parenthesis squared plus left parenthesis negative 10 right parenthesis squared plus left parenthesis 4 right parenthesis squared end root space equals space square root of 180 therefore space space space area space of space vertical line vertical line space gm 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 180

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