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Find the all the values of lamda such th...

Find the all the values of lamda such that (x,y,z)!=(0,0,0)`and `x(hati+hatj+3hatk)+y(3hati-3hatj+hatk)+z(-4hati+5hatj)=lamda(xhati+yhatj+zhatk)`

Text Solution

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The correct Answer is:
`0, -1`

`(hati + hatj +3hatk)x + (3hati-3hatj +hatk)y+ (-4hati+5hatj)z`
`= lamda (x hati +yhatj+zhatk)`
Comparing coefficient of `hati, x + 3y -4z=lamda x`
`rArr (1-lamda)x + 3y- 4z =0" "` (i)
Comparing coefficient of `hatj, x-3y+5z= lamday`
`rAtr x - (3+lamda)y + 5z =0" " `(ii)
Comparing coefficient of `hatk, 3x+y+0z =lamda z`
`rArr 3x+y -lamda z =0" "` (iii)
All the above three equations are satisfied for x, y and z not all zero if
`|{:(1-lamda,,3,,-4),(1,,-(3+lamda),,5),(3,,1,,-lamda):}| =0`
or `" "(1-lamda)[3lamda +lamda^(2)-5]-3[-lamda -15]- 4[1+9+3 lamda] =0`
or `" " lamda^(3) + 2lamda^(2) +lamda =0`
or `" " lamda(lamda + 1)^(2) =0`
or `" " lamda =0, -1`
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Knowledge Check

  • The value of lambda in R such that (x, y, z) ne (0, 0, ) and (2hati+3hatj-4hatk)x+(3hati-hatj+2hatk)y+(i-2hatj)z=lambda(xhati+yhatj+zhatk) lies in

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