Decomposition of a vector in a plane

Theorem

and

be any two non-zero and non-collinear vectors then any vector

in the plane of

and

can be uniquely expressed as the sum of two vectors parallel to the vectors

and

. Proof: Let

and

be any two non-zero and non-collinear plane vectors. Let

be any other vector in the plane of the vectors

and

. Now from the point P draw the straight lines PG and PH parallel to the lines OA and OB as shown in the succeeding figure below.