Dot product of two vectors is defined as the product of their magnitudes and cosine of the angle between them. 

                 

Now          

                       

Hence, dot product of two vectors is commutative in nature. 

The characteristics of dot product are: 

(i)  Dot product of two vectors is commutative.

Mathematically,    

(ii)  Dot product is distributive. 

Mathematically,   

(iii) Dot product of two perpendiculars vectors is zero.

Mathematically              if    

(iv) Dot product of vector with itself is equal to square of magnitude of vector. 

Mathematically,    

Let  be the angle between the given two vectors.

Thus, 
                         

Here,                      

                               

and           

∴      , is the angle between the vectors. 

When the magnitude of a vector is zero, it is known as a zero vector. Zero vector has an arbitrary direction. 

Examples: (i) Position vector of origin is zero vector.

(ii) If a particle is at rest then displacement of the particle is zero vector. 

(iii) Acceleration of uniform motion is zero vector.