Example – 5
Let  1 ,  and 
be non-coplanar vectors. Are the vectors 2 - + 3, + - 2 and + - 3 coplanar or
non-coplanar?
Solution: Three vectors are coplanar if there exist scalars l,m Î R using which one vector can be expressed as the linear combination of the other two.
Let us try to find such scalars:
Since , , are non-coplanar, we must have
2 -l -m = 0
-1-l -m = 0
3+ 2l + 3m = 0
This system, as can be easily verified , does not have a solution for l and m.
Thus, we cannot find scalars for which one vector can be expressed as the linear combination of the other two, implying the three vectors must be non-coplanar.
As an additional exercise, show that for three non-coplanar vectors , and , the vectors
- 2 + 3, - 3 + 5
and -2 -3 - 4
are coplanar.
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