Prove that three vectors `vec(a),vec(b)andvec(c)` are coplanar if and only if there exists non-zero linear combination `xvec(a)+yvec(b)+zvec(c)=vec(0)`.

Prove that three vectors `vec(a),vec(b)andvec(c)` are coplanar if and

1.	Prove that three vectors `vec(a),vec(b)andvec(c)` are coplanar if and only if there exists non-zero linear combination `xvec(a)+yvec(b)+zvec(c)=vec(0)`.
Answer» Let `vec(a),vec(b)andvec(c)` be coplanar. Case I : Let `vec(a) andvec(b)` are collinear. Then there exists scalars x, y at least one of them is non-zero such that `xvec(a)+yvec(b)=0`. `:.xvec(a)+yvec(b)+zvec(c)=0` is a linear combination where z=0 Case II : Suppose none of the vectors `vec(a),vec(b)andvec(c)` are collinear. AS `vec(c` is coplanar with `vec(a)andvec(b)`, there exists scalars x, y such that `vec(c)=xvec(a)+yvec(b)` `:.xvec(a)+yvec(b)+zvec(c)=0` is the required non-zero linear combination, where z=-1. Conversely, suppose `xvec(a)+yvec(b)+zvec(c)=0`, where one of x, y, z is non-zero say `zne0`. `:.vec(c)=(x)/(z)vec(a)+(y)/(z)vec(b)` `:.vec(c)` is coplanar with `vec(a)andvec(b)` `:.vec(a),vec(b)andvec(c)` are coplanar vectors.

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Prove that three vectors `vec(a),vec(b)andvec(c)` are coplanar if and only if there exists non-zero linear combination `xvec(a)+yvec(b)+zvec(c)=vec(0)`.

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