# Product Of Two Vectors Part 1 (F.Sc-Physics-Chapter 2)

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**)

Or

**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?**

You are falling off the edge. what should you do to avoid falling?

**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