Operations with Vectors

Vector Addition:
 Consider that a man is walking on a path ABC (from A to B and then from B to C) as shown in the figure. In this case displacement AC vector can be written as the sum of the AB vector and BC vector.
                                AB + BC = AC 
                 this is known as the triangle rule of vectors. 

                                  
So, in the general case if we have two vectors a and b are positioned so that the initial point of one coincides with the terminal point of the other. Then, vector (a+b) represented by the third side c vector of the triangle, which is the addition of (a+b) vector  or we can say resultant vector of a vector and b vector.
                                         a + b = c  
If we rotate b vector from π angle without changing its magnitude then according to figure in triangle ∆ABC'
                                           a + (-b) = c'
                                           a - b = c'  

       Consider that OA vector and OB vector are two vectors in space. Draw a AC vector parallel to OB vector so that |AC| = |OB| and similarly BC vector parallel to OA vector so that |BC| = |OA|

   
               In parallel quadrilateral OACB.
                 So, according to triangle rule of vectors in ∆OAC

                                   OA + AC = OC

                             ∵   |AC| =|OB|

                             ∴   OA + OB = OC

Polygon law of Vector Addition:

       According to polygon law of vector addition if a number of vectors can be represented in magnitude and direction by the sides of a polygon taken in the same order, then their resultant or addition is represented in magnitude and direction by the closing side of the polygon taken in the opposite order.

      If we have a vector, b vector, c vector and d vector as shown in figure then their resultant vector = a + b + c + d

      

• Vector addition shows following properties -       

1. Commutative: a (vector) + b (vector) = b (vector) + a (vector)
2. Associativity: (a + b) + c = a + (b + c)
3.   a + 0(vector)  = a = 0(vector) + a
4.   a + (-a) = 0(vector) = (-a) + a 

                      NOTE:- Here AB, BC, AC, AC', a, b, c, c', 0, OA, OB, OC are vectors.

Multiplication of Vectors by Scalar: 

     Multiplication of scalar quantity with vector changes magnitude of the vector. Consider a vector ⊽(xî, yĵ, zk̂) , which multiplied by a scalar λ -
                        ∵   ⊽ = xî + yĵ + zk̂
                        ∴  λ.⊽ = λxî + λyĵ + λzk̂
     Now, magnitude of resultant vector is λ times of magnitude of ⊽. If λ is positive then the direction of the vector will remain same and if λ is negative then the direction of vector becomes opposite(π) of the given vector.

     example: If  Ā = 3î - 7ĵ + 6k̂ . Find the following vectors-
                      (i) 5Ā     (ii) -1/3(Ā)    
     Solution: 
                   (i) 5Ā = 5(3î - 7ĵ + 6k̂) = 15î - 35ĵ + 30k̂
                     (ii) -1/3(Ā) = -1/3(3î - 7ĵ + 6k̂) = -î + 7/3ĵ - 2k̂

Scalar Product of two Vectors:

       If A and B are two vectors and θ is the angle between them then scalar product of A vector and B vector can be written as -

                     A.B = |A||B|cosθ

                                                 here, 0 ≤ θ ≤ π

Properties:

1.  A.A = |A||A|cos(0) =|A||A| = |A|²      (A is co linear vector of same magnitude)
2. A.B =B.A (commutative)
3. A.(B + C) = A.B + A.C (distributive)
4. if  A⊥B then cos(π/2) = 0   ⇒  A.B = 0     (A and B are               perpendicular to each other)
5.   î.î = ĵ.ĵ = k̂.k̂ = 1
6.   î.ĵ = ĵ.k̂ = k̂.î

Projection Vector:

             Projection of A vector on B vector = |A|cosθ

                                                                     = {|A||B|cosθ}/|B|

                                                                     = A.B/|B|

               NOTE:- Here A and B are vectors.     

   example: 

        

Vector Product of two Vectors:
         If A and B are two vectors and θ is the angle between them then vector product of A vector and B vector can be written as -
                             A🗙B = |A||B|sinθ 
                                                 here, 0 ≤ θ ≤ π

Properties:

1.    If a vector and b vector are parallel to each other or co-linear     then,    a🗙b = 0    (∵ θ = 0)
2.    a🗙b ≠ b🗙a   (not commutative)
3.    a🗙(b + c) = a🗙b + a🗙c    (distributive)
4.    î🗙î = ĵ🗙ĵ = k̂🗙k̂ = 0
5.    î🗙ĵ = k̂ , ĵ🗙k̂ = î ,  k̂🗙î = ĵ
6.    î🗙k̂ = -ĵ , k̂🗙ĵ = - î , ĵ🗙î = -k̂