Prove that three vectors a,b and c are coplanar, if and only if, there exists a non-zero linear combination xa+yb +zc=0 - Mathematics and Statistics

Prove that three vectors bara, barb and barc  are coplanar, if and only if, there exists a non-zero linear combination xbara+ybarb +z barc=0

Solution

Let bara, barb, barc  be coplanar vectors. Then any one of them, say bara , will be the linear combination of  bar b and bar c.

there exist scalars α and β such that

bara=alpha barb +beta barc

therefore (-1)bara+alpha barb+beta barc=bar0

i.e " "x bara+y barb+z barc=bar0

Let x ≠ 0, then divide (1) by x, we get,

i.e" " bara+(y/x)barb+(z/x)barc=bar0

therefore bara=(-y/x)barb+(-z/x)barc

i.e. bara=alpha barb+beta barc, "where " alpha=(-y/x) and beta=-z/x are scalar

therefore bara  is the linear combination of  bar b and barc.

Hence, bara, barb, barc are coplanar.

Concept: Vector and Cartesian Equations of a Line - Conditions of Coplanarity of Three Vectors
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