Let a^ and b^  be mutually perpendicular unit vectors. Then for any arbitrary   r→

Let r→=x1a^+x2b^+x3(a^×b^)⇒ r→⋅a^=x1+x2a^⋅b^+x3a^⋅(a^×b^)=x1 Also, r→⋅b^=x1a^⋅b^+x2+x3b^⋅(a^+b^)=x2 and r→⋅(a^×b^)=x1a^⋅(a^×b^)+x2b^⋅(a^×b^)+x3(a^×b^)⋅(a^×b^)=x3 ⇒ r→=(r→⋅a^)a^+(r→⋅b^)b^+(r→⋅a^×b^))(a^×b^)