Question

Show that the vectors `2veca-vecb+3vecc, veca+vecb-2vecc and veca+vecb-3vecc` are non-coplanar vectors (where `veca, vecb, vecc` are non-coplanar vectors).

Answer 1

Consider `2veca-vecb+3vecc=x (veca+vecb-2vecc) + y(veca+vecb-3vecc)`
or `" "2veca-vecb+3vecc= (x+y)veca+ (x+y)vecb+ (-2x-3y)vecc`
`" "x+y=2" "` (i)
`" "x+y=-1" "` (ii)
`" "-2x-3y=3" "` (iii)
Multiplying (i) by 3 and adding it to (iii), we get
`x=9`
From (i), `9+y=2 or y =-7`
Now putting `x=9 and y=-7` in (ii), we get
`" "9-7=-1`
or `2=-1`, which is not true.
Therefore, the given vectors are not coplanar.
Alternate method :
We have determinant of co-efficients as
`" "|{:(2,,-1,,3),(1,,1,,-2),(1,,1,,-3):}| = -3 ne 0`
Hence, vectors are non-coplanar.