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).
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.
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
- 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.
- 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)
- 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,
what should you do?
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