Given that,

                         

Let angle between  be  and that between 

∴     

      

                

Now,    if   then A = C, and

              then 

Thus A is not necessarily equal to C.

Resultant of  is

 
∴   Unit vector in direction of 

Magnitude of vector A is,

     

Unit vector in direction of vector 



The vector that has magnitude same as that of vector and parallel to vector 



      

It is given that angle between  is  

Therefore the angle between  will be 

Here,        

∴         

               


               .

Hence proved. 

By using distributive and commutative law, the dot product can be evaluated as, 

   

     
 

That is,