Question

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`

Answer 1

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.