Prove that three vectors `veca, vecb and vec c` are coplanar, if and only if, there exists a non-zero linear combination `xbara+ybarb+zbarc=baro .`

Explanation

`veca, becb, vecc` be coplanar vectors. Then any one of them, say `veca` will the linear combination of `vecb` and `vecc`.
`therefore ` There exist scalars `alpha and beta` such that
`veca=alpha vecb+beta vecc`
`therefore (-1)veca+alpha vecb+beta vecc=0`
Where `x=-01, y=alpha, z=beta` when are not all zero simultaneously then
`xvec a+y vecb+z vecc=0`
Conversely : Let there exist scalars x,y, z not all zero such that
`xveca +yvecb+z vecc=0" ...(i)"`
Let `x ne 0`, then divides (i) by x, we get
`x[veca(y)/(x)vecb+(z)/(x)vecc]=0`
`veca+(y)/(x)vecb+(z)/(x)vecc=0`
Hence x, y, z are not zero scalars. Hence Proved.