Dot product of two vectors is defined as the product of their magnitudes and cosine of the angle between them.Â
         Â
Now      Â
            Â
Hence, dot product of two vectors is commutative in nature.Â
Find the angle between the vectors  andÂ
Let  be the angle between the given two vectors.
Thus,Â
            Â
Here, Â Â Â Â Â Â Â Â Â Â Â
               Â
and      Â
∴    , is the angle between the vectors.Â
What are the characteristics of dot product?
The characteristics of dot product are:Â
(i) Â Dot product of two vectors is commutative.
Mathematically, Â Â Â
(ii) Â Dot product is distributive.Â
Mathematically, Â Â
(iii) Dot product of two perpendiculars vectors is zero.
Mathematically        if   Â
(iv) Dot product of vector with itself is equal to square of magnitude of vector.Â
Mathematically, Â Â Â
When the magnitude of a vector is zero, it is known as a zero vector. Zero vector has an arbitrary direction.Â
Examples: (i) Position vector of origin is zero vector.
(ii) If a particle is at rest then displacement of the particle is zero vector.Â
(iii) Acceleration of uniform motion is zero vector.