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Vector Algebra

Question
CBSEENMA12034170
Wired Faculty App

If 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 2 space straight k with hat on top comma space space straight b with rightwards arrow on top space equals space 2 space straight i with hat on top space minus space 3 space straight j with hat on top space minus space 4 space straight k with hat on top comma then show that straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space equals space minus straight b with rightwards arrow on top space cross times space straight a with rightwards arrow on top

Solution

Here,    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 plus space 2 space straight k with hat on top comma space space straight b with rightwards arrow on top space equals space 2 space straight i with hat on top space minus space 3 space straight j with hat on top space minus space 4 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 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 2 row 2 cell negative 3 end cell cell negative 4 end cell 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 2 row cell negative 3 end cell cell negative 4 end cell end table close vertical bar space minus space straight j with hat on top space open vertical bar table row 1 2 row 2 cell negative 4 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 2 3 end table close vertical bar space space space space space equals left parenthesis 4 plus 6 right parenthesis space straight i with hat on top space minus space left parenthesis negative 4 minus 4 right parenthesis space straight j with hat on top space plus space left parenthesis negative 3 plus 2 right parenthesis straight j with hat on top therefore space space space straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space equals space 10 space straight i with hat on top space plus space 8 space straight j with hat on top space minus space straight k with hat on top space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis space space space
Again,   straight b with rightwards arrow on top space cross times space straight a 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 cell negative 3 end cell cell negative 4 end cell row 1 cell negative 1 end cell 2 end table close vertical bar space equals space straight i with hat on top space open vertical bar table row cell negative 3 end cell cell negative 4 end cell row cell negative 1 end cell 2 end table close vertical bar space minus space straight j with hat on top space open vertical bar table row 2 cell negative 4 end cell row 1 2 end table close vertical bar space plus space straight k with hat on top space open vertical bar table row 2 cell negative 3 end cell row 1 1 end table close vertical bar
                      equals left parenthesis negative 6 minus 4 right parenthesis space straight i with hat on top space minus space left parenthesis 4 plus 4 right parenthesis space straight j with hat on top space plus space left parenthesis negative 2 plus 3 right parenthesis space straight k with hat on top equals space minus 10 space straight i with hat on top space minus space 8 space straight j with hat on top space plus space straight k with hat on top space equals space minus left parenthesis 10 straight i with hat on top plus space 8 straight j with hat on top space minus space straight k with hat on top right parenthesis space equals space minus left parenthesis straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top right parenthesis space space space space left square bracket because space of space left parenthesis 1 right parenthesis right square bracket therefore space straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space equals space minus straight b with rightwards arrow on top space cross times space straight a with rightwards arrow on top

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