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Vector Algebra

Question
CBSEENMA12034191
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Find the value of straight lambda so that the vectors 2 space straight i with hat on top space plus 4 space straight j with hat on top space minus space 2 space straight k with hat on top and 3 straight i with hat on top space plus space 6 straight j with hat on top space plus space straight lambda straight k with hat on top space are (i) parallel (ii) perpendicular to each other.

Solution

Let straight a with rightwards arrow on top space equals space 2 space straight i with hat on top space plus space 4 space straight j with hat on top minus space 2 space straight k with hat on top comma space space straight b with rightwards arrow on top space equals space 3 space straight i with hat on top plus 6 space straight j with hat on top space plus space straight lambda 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 4 cell negative 2 end cell row 3 6 cell space space straight lambda end cell end table close vertical bar space equals space straight i with hat on top space open vertical bar table row 4 cell negative 2 end cell row 6 cell space space straight lambda end cell end table close vertical bar space minus space straight j with hat on top space open vertical bar table row 2 cell negative 2 end cell row 3 cell space space space straight lambda end cell end table close vertical bar space plus space straight k with hat on top space open vertical bar table row 2 4 row 3 6 end table close vertical bar space space space space space space space space space space space space equals space left parenthesis 4 straight lambda plus 12 right parenthesis space straight i with hat on top space minus space left parenthesis 2 straight lambda plus 6 right parenthesis straight j with hat on top space plus space left parenthesis 12 minus 12 right parenthesis straight k with hat on top space equals space left parenthesis 4 straight lambda plus 12 right parenthesis straight i with hat on top space minus space left parenthesis 2 straight lambda plus 6 right parenthesis straight j with hat on top Since space straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space are space parallel 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 0 with rightwards arrow on top therefore space space space space left parenthesis 4 space straight lambda space plus 12 right parenthesis space straight i with hat on top space minus space left parenthesis 2 straight lambda plus 6 right parenthesis straight j with hat on top space equals 0 with rightwards arrow on top rightwards double arrow space space 4 straight lambda plus 12 space equals space 0 comma space space 2 straight lambda plus 6 space equals space 0 space space space rightwards double arrow space space space space straight lambda space equals space minus 3
(ii) Since straight a with rightwards arrow on top space and space straight b with rightwards arrow on top are perpendicular
   therefore space space straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space 0 therefore space space space left parenthesis 2 right parenthesis thin space left parenthesis 3 right parenthesis space plus space left parenthesis 4 right parenthesis thin space left parenthesis 6 right parenthesis space plus space left parenthesis negative 2 right parenthesis space straight lambda space equals space 0 therefore space space space 6 plus 24 minus 2 straight lambda space equals space 0 space space space space space space space space space space space space space space space space space space space rightwards double arrow space 2 space straight lambda space equals 30 space space space space space space space space space space space space space space space space rightwards double arrow space space space space straight lambda space equals space 15

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