If [math]a^{\rightarrow}[/math] and [math]b^{^{\rightarrow}}[/math] be any two non-zero and non-collinear vectors then any vector [math]r^{\rightarrow}[/math] in the plane of [math]a^{\rightarrow}[/math] and [math]b^{\rightarrow}[/math] can be uniquely expressed as the sum of two vectors parallel to the vectors [math]a^{\rightarrow}[/math] and [math]b^{\rightarrow}[/math].[br][b]Proof:[/b] Let [math]OA^{\rightarrow}=a^{\rightarrow}[/math] and [math]OB^{\rightarrow}=^{ }b^{\rightarrow}[/math] be any two non-zero and non-collinear plane vectors. Let [math]OP^{\rightarrow}=r^{\rightarrow}[/math]be any other vector in the plane of the vectors [math]a^{\rightarrow}[/math] and [math]b^{\rightarrow}[/math]. 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.