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Vector Algebra

Short Answer Type

261. Find the area of the parallelogram whose adjacent sides are determined by the vectors $straight a with rightwards arrow on top space equals straight i with hat on top space minus straight j with hat on top plus 3 space straight k with hat on top$ and $straight b with rightwards arrow on top space equals space 2 space straight i with hat on top space minus space 7 space straight j with hat on top space plus space straight k with hat on top.$

Here $straight a with rightwards arrow on top space equals straight i with hat on top space minus straight j with hat on top space plus space 3 space straight k with hat on top comma space space space space straight b with rightwards arrow on top space equals space 2 space straight i with hat on top space minus space 7 space straight j with hat on top space plus space straight k with hat on top$
$straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space equals space open vertical bar table row cell straight i with hat on top end cell cell straight j with hat on top end cell cell straight k with hat on top end cell row 1 cell negative 1 end cell 3 row 2 cell negative 7 end cell 1 end table close vertical bar space equals space straight i with hat on top space open vertical bar table row cell negative 1 end cell 3 row cell negative 7 end cell 1 end table close vertical bar space minus space straight j with hat on top space open vertical bar table row 1 3 row 2 1 end table close vertical bar plus straight k with hat on top space open vertical bar table row 1 cell negative 1 end cell row 2 cell negative 7 end cell end table close vertical bar space space space equals space left parenthesis negative 1 plus 21 right parenthesis space straight i with hat on top space minus space left parenthesis 1 minus 6 right parenthesis space straight j with hat on top space plus space left parenthesis negative 7 plus 2 right parenthesis space straight k with hat on top space equals space 20 space straight i with hat on top space plus space space 5 space stack straight j space with hat on top space minus space 5 space straight k with hat on top therefore space space space space open vertical bar straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top close vertical bar space equals space square root of 400 plus 25 plus 25 end root space equals square root of 450 space equals space square root of 255 space cross times 2 end root space equals space 15 square root of 2 therefore space space space space area space of space parallelogram space equals space 15 square root of 2 space sq. space units. space$