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

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

|A| is the modulus, or magnitude of A, |B| is the modulus of B, and θ is the angle between A and B. 

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

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

A.B=A (magnitude of the component of B in the direction of A) = A(B * cos θ) = AB * cos θ

SimilarlyB. 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 8 with the line of action of F, as shown in Fig. 2.11.

Work done = (effective component of force in the directionof motion) x distance moved= (F * cos θ) * d = Fd * cos θ

Using vector notation

F. d = F * d * cos θ = work * done

Applications of Scalar Product

There are numerous applications in vector theory, some of them include:

• Projection of a Vector: A scalar product is used to determine the projection of a vector onto another vector. The projection of vector a onto vector b is given by A.B/|B|. Similarly, the projection of vector b onto vector a is given by A.B/|A|.
• Scalar Triple Product: The scalar product is used to calculate the scalar triple product of three vectors. The formula for the scalar triple product is a.(b × c) = b.(c × a) = c.(a × b)
• Angle Between Two Vectors: The scalar product determines the angle between two vectors using the formula cos θ = (A.B)/(|A| |B|).

Now, that we have understood the concept of scalar product, let us go through some of the important properties of the scalar product of vectors A and B that will help us in solving various problems:

• Commutative Property - Scalar product is commutative, that is, A.B = B.A

  Since A.B = AB cos θ and B. A = BA  cos θ hence, A .B=B.A

• Distributive Property - The scalar product of vectors follows the distributive property:
  a.(b + c) = a.b + a.c
  (a + b).c = a.c + b.c
  a.(b - c) = a.b - a.c
  (a - b).c = a.c - b.c

• The scalar product of two mutually perpendicular vectors is zero. A. B = AB * cos 90 deg = 0
  • Scalar product of two parallel. vectors are equal to the product of their magnitudes.
  • The self-product of a vector is equal to the square of its magnitude.

$({�}_1\overrightarrow{�}).({�}_2\overrightarrow{�})={�}_1{�}_2(\overrightarrow{�}.\overrightarrow{�})$

• The angle between the vectors  θ=cos^ -1 [ vec A * vec B /AB ]
• The scalar product of two vectors will be maximum when cos  θ = 1, i.e.  θ = 0 deg, i.e., when the vectors are parallel;( vec A . vec B ) max = AB
• The scalar product of two vectors will be minimum, when cos θ = - 1, i.e. θ = 180 deg.
• Associative Property:

      If a is a vector and  c, and d are scalars then,

      c (da) = (cd) a

  4. Identity Property:

      If a is a vector then,

      1⋅a = a

  5. Multiplicative Property of 0:

     If a is a vector then,

     0 (a)=0

^i⋅^j=|^i||^j|cos90°=(1)(1)(0)=0,^i⋅^k=|^i||^k|cos90°=(1)(1)(0)=0,^k⋅^j=|^k||^j|cos90°=(1)(1)(0)=0

^i⋅^i=i2=^j⋅^j=j2=^k⋅^k=k2=1.

$\overrightarrow{A}=A_x\hat{i}+A_y\hat{j}+A_z\hat{k}\text{and}\overrightarrow{B}=B_x\hat{i}+B_y\hat{j}+B_z\hat{k},$