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Vector Algebra

Question
CBSEENMA12034211
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Find the area of a parallelogram whose adjacent sides are given by the vectors 
straight a with rightwards arrow on top space equals 3 straight i with hat on top space plus straight j with hat on top space plus space 4 space straight k with hat on top and straight b with rightwards arrow on top space equals space straight i with hat on top space minus space straight j with hat on top space plus space straight k with hat on top.

Solution

Here 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 plus space 4 space straight k with hat on top comma space space straight b with rightwards arrow on top space equals space straight i with hat on top space minus 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 3 1 4 row 1 cell negative 1 end cell 1 end table close vertical bar space equals space straight i with hat on top space open vertical bar table row cell space space 1 end cell 4 row cell negative 1 end cell 1 end table close vertical bar space minus space straight j with hat on top space open vertical bar table row 3 4 row 1 1 end table close vertical bar space plus space straight k with hat on top space open vertical bar table row 3 cell space space space 1 end cell row 1 cell negative 1 end cell end table close vertical bar space space space space space space space space space space space space space equals space left parenthesis 1 plus 4 right parenthesis space straight i with hat on top space minus space left parenthesis 3 minus 4 right parenthesis space straight j with hat on top plus space left parenthesis negative 3 minus 1 right parenthesis space straight k with hat on top space equals space 5 straight i with hat on top space space plus straight j with hat on top space minus space 4 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 left parenthesis 5 right parenthesis squared plus left parenthesis 1 right parenthesis squared plus left parenthesis negative 4 right parenthesis squared end root space equals space square root of 25 plus 1 plus 16 end root space equals space square root of 42 therefore space space area space of space parallelogram space equals space square root of 42 space space sq. space units. space

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