There are two types of vector multiplications. The product of these two types are known as scalar product of two vector quantities is a scalar quantity, while vector product of two vector quantities is a vector quantity.

Scalar or Dot Product

The scalar product of two vectors A and B is written as A.B and is defined as

A.B = AB cos θ                   ……………………                     (2.17)

Where A and B are the magnitudes of vectors A and B and θ is the angle between them.

For physical interpretation of dot product of two vectors A and B, these are first brought to a common origin (fig. 2.10 a).


Fig. 2.10 (a)


Fig. 2.10 (b)

Then, A.B = (A) (projection of B on A)


A.B = A (magnitude of component of B in the direction of A)

= A (B cos θ) = AB cos θ

Similarly               B.A = B (A cos θ) = BA cos θ

We come across this type of product when we consider the work done by a force F whose point of application moves a distance d in a direction making an angle θ with the line of action of F, as shown in Fig. 2.11.


Fig. 2.11

Work done = (effective component of force in the direction of motion) x distance moved

= (F cos θ) d = Fd cos θ

Using vector notation

F.d = Fd cos θ = work done

Characteristics of scalar product

  1. Since A.B = AB cos θ and B.A = BA cos θ, hence, A.B = B.A the order of multiplication is irrelevant. In other words. Scalar product is commutative.
  2. The scalar product of two mutually perpendicular vectors is zero. A.B = AB cos90⁰ = 0

In case of unit vectors î , ĵ and k̂, since they are mutually perpendicular, therefore,

Î . ĵ = ĵ . k̂ = î = 0                  ……………               (2.18)

  1. The scalar product of two parallel vectors is equal to the product of their magnitudes. Thus for parallel vectors (θ = 0⁰)

A.B = AB cos 0⁰ = AB

In case of unit vectors

Î . Î = ĵ. ĵ = k̂. k̂ = 1              ……………..             (2.19)

And for antiparallel vectors (θ = 180⁰)

A.B = AB cos 180⁰ = -AB

4.            the self product of a vector A is equal to square of its magnitude.


5.            scalar product of two vectors A and B in terms of their rectangular components


Equation 2.17 can be used to find the angle between two vectors: since,


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Example 2.4: A force F = 2 î + 3ĵ units, has its point of application moved from point A (1,3) to the point B (5,7) find the work done.


Work done = F.d = (2î + 3ĵ). (4î + 4ĵ)

= 8 + 12 = 20 units

Example 2.5: find the projection of vector A=2î -8ĵ +k̂ in the direction of the vector B = 3î – 4ĵ – 12k̂.

Solution: if θ is the angle between A and B, then A cosθ is the required projection.


Where B̂ is the unit vector in the direction of B


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