The number of distinct real values of , for which the vectors - ^2 i + j + k , i - ^2 j + k and i + j - ^2 k are coplanar, is

Step-by-step solution

For co-planar vectors, we have aligned & | array ccc - ^2 & 1 & 1 1 & - ^2 & 1 1 & 1 & - ^2 array |=0 & - ^2 ( ^4-1 )-1 (- ^2-1 )+1 (1+ ^2 )=0 & - ^6+ ^2+ ^2+1+1+ ^2=0 & ^6-3 ^2-2=0 & t^3-3 t-2=0 & [ For t = ^2 ] & ( t +1) ( t ^2- t -2 )=0 & ( t +1)( t +1)( t -2)=0 & t=-1 or t=2 & i.e., ^2=-1 or ^2=2 aligned for real values of , we get = 2 , two values.