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)`

Explanation

Here,
`(hati+hatj+hatk)x+(3hati-3hatj+hatk)y+(-4hati+5hatj)z=lamda(xhati+yhatj+zhatk)`
On comparing the coefficient of `hati,hatj and hatk`, we get
`x+3y-4z=lamdax`
`implies(1-lamda)x+3y-4z=0` . . . (i)
`x-3y+5z=lamda y`
`implies x-(3+lamda)y+5z=0` . . (ii)
`3x+y=lamdaz`
`implies 3x+y-lamdaz=0` . . . (iii)
The eqs. (i), (ii) and (iiI) will have a non-trivial solution, if
`|(1-lamda,3,-4),(1,-(3+lamda),5),(3,1,-lamda)|=0`
`[because (x,y,z)ne(0,0,0)thereforeDelta=0]`
`implies(1-lamda){lamda(3+lamda)-5}-3{-lamda-15}-4{1+3(lamda+3)}=0`
`implies(1-lamda){lamda^(2)+3lamda-5}-3{-lamda-15}-4{3lamda+10}=0`
`implies lamda^(3)+2lamda^(2)+lamda=0`
`implies lamda(lamda^(2)+2lamda+1)=0`
`implies lamda(lamda+1)^(2)=0`
`therefore lamda =0` or `lamda=-1`.