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).

Show that the vectors `2veca-vecb+3vecc, veca+vecb-2vecc and veca+vecb

1.	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» 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.

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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).

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