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Vector Algebra

Question
CBSEENMA12034188
Wired Faculty App

Find the value of straight lambda so that the two vectors 2 space straight i with hat on top space minus space 4 space straight j with hat on top space plus space straight k with hat on top space space and space 4 space straight i with hat on top space minus space 8 space straight j with hat on top space plus space straight lambda space straight k with hat on top 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 minus space 4 space straight j with hat on top space plus space straight k with hat on top comma space space space straight b with rightwards arrow on top space equals space 4 space straight i with hat on top space minus space 8 space straight j with hat on top space plus space straight lambda space straight k with hat on top
(i)  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 cell negative 4 end cell 1 row 4 cell negative 8 end cell straight lambda end table close vertical bar space equals space straight i with hat on top space open vertical bar table row cell negative 4 end cell 1 row cell negative 8 end cell straight lambda end table close vertical bar space minus straight j with hat on top space open vertical bar table row 2 cell negative 1 end cell row 4 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 cell negative 4 end cell row 4 cell negative 8 end cell end table close vertical bar
                  equals space left parenthesis negative 4 straight lambda plus 8 right parenthesis space straight i with hat on top space minus space left parenthesis 2 straight lambda minus 4 right parenthesis space straight j with hat on top space plus space straight k with hat on top space left parenthesis negative 16 plus 16 right parenthesis space equals space left parenthesis negative 4 space straight lambda space plus 8 right parenthesis straight i with hat on top minus left parenthesis 2 straight lambda minus 4 right parenthesis straight j with hat on top
Since straight a with rightwards arrow on top space and space straight b with rightwards arrow on top space are space parallel
 therefore 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 left parenthesis negative 4 space straight lambda space plus space 8 right parenthesis space straight i with hat on top space minus space left parenthesis 2 space straight lambda space minus space 4 right parenthesis space straight j with hat on top space equals space 0 with rightwards arrow on top rightwards double arrow space space space minus 4 straight lambda plus 8 space equals space 0 comma space space space 2 straight lambda minus 4 space equals space 0 space space space space space space rightwards double arrow space space space straight lambda space equals space 2
(ii) Since straight a with rightwards arrow on top space and space straight b with rightwards arrow on top are perpendicular   
therefore space space space space space straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space 0 therefore space space left parenthesis 2 right parenthesis thin space left parenthesis 4 right parenthesis space plus space left parenthesis negative 4 right parenthesis thin space left parenthesis negative 8 right parenthesis space plus space left parenthesis 1 right parenthesis space straight lambda space equals space 0 therefore space space 8 plus 32 plus straight lambda space equals space 0 space space space space space space space space space space space space space space space space space space rightwards double arrow space space space straight lambda space equals space minus 40

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