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)`

Answer

`0, -1`

Explanation

`(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`