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

   

     
 

That is,

     

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 



      

Resultant of  is

 
∴   Unit vector in direction of 

It is given that angle between  is  

Therefore the angle between  will be 

Here,        

∴         

               


               .

Hence proved. 

Given that,

                         

Let angle between  be  and that between 

∴     

      

                

Now,    if   then A = C, and

              then 

Thus A is not necessarily equal to C.