Dot products of vectors
When two vectors are multiplied such that the product is a scalar quantity, it is called DOT product or Scalar product.
The scalar product or dot product of any two vectors 
and 
, denoted as .(read dot) is defined as the product of their magnitude with cosine of angle between them.
Thus, . = AB cos θ
{where θ is the angle between the vectors
Properties of Dot Product
- The dot product of two vectors is always a scalar quantity. It is positive, if angle between the vectors is acute (i.e. < 90º) and it is negative, if angle between them is obtuse (i.e.90 < θ ≤ 180º)
2. It is commutative, i.e. 
3. It is distributive, i.e. 
4. Scalar product of two vectors will be maximum when cos θ = max = 1, i.e. θ = 0º, i.e. vectors are parallel.
5. If the scalar product of two nonzero vectors vanishes then the vectors are orthogonal i.e θ =90º
6. The scalar product of a vectors by itself is termed as self dot product and is given by
7. In case of unit vector 
8. In case of orthogonal units 
9. If vectors are represented in terms of their components, then
10. As by definition 
Or 
12. Since, 
Geometrically, B cosθ is the projection (component) of onto and A cosθ is the projection of onto as shown. So is the product of the magnitude of and the component of along and vice versa.
Magnitude of the component of along 
Component of along
Similarly, Component of along 
Dot Product Of Vectors Video
(In Hindi + English Mix)
Example: If the vectors 
and 
are perpendicular to each other. Find the value of ‘a’?
Solution. If vectors 
and 
are perpendicular
Example: Find the component of 
along 
Solution. Magnitude of Component of along is given by 
Example. Find angle between 
Solution. We have cos θ = 
