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Addition and Subtraction of Vectors



Geometrical method 
                         geometrical-method 

To find a  + b , shift  vector b  such that its initial point coincides with the terminal point of vector a. Now, the vector whose initial point coincides with the initial point of vector a , and terminal point coincides with the terminal point of vector b  represents (a +b ) as shown in the above figure. 

To find (b +a ), shift a  such that its initial point coincides with the terminal point b . A vector whose initial point coincides with the initial point of b  and terminal point coincides with the terminal point of a  represents (b +a ).

                       addition-of-two-vectors 

Law of Parallelogram of Vectors 

The addition of two vectors may also be understood by the law of parallelogram. 

According to this law if two vectors P and Q are represented by two adjacent sides of a parallelogram both pointing outwards as shown in the figure below , then the diagonal drawn through the intersection of the two vectors represents the resultant (i.e. vector sum of P and Q). If Q is displacement from position AD to BC by displacing it parallel to itself, this method becomes equivalent to the triangle method. 

                              parallelogram-method 

In case of addition of two vectors by parallelogram method as shown in figure, the magnitude of resultant will be given by, 

                                       (AC)2 = (AE)2 + (EC)2 

                               or R2 = (P + Q cos θ)2 (Q sin θ)2 


Addition and Subtraction of Vectors

                                                     
                                 
                                   or R = √(P2+ Q2 )+ 2PQcos θ 

                    And the direction of resultant from vector P will be given by
 
                              tanǾ = CE/AE = Qsinθ/(P+Qcosθ) 

                                   Ǿ=tan-1 [Qsinθ/(P+Qcosθ)]  

Magnitude and direction of the resultant 

 The magnitude of resultant will be 

                          R = √(P2+ P2+2P2 cos"α" ) = 2Pcosα/2 Ans.

The direction of the resultant is 

                    Ǿ = tan-1 [Psinθ/(P+Pcosθ)] = tan-1tan(θ/2) = Ǿ = θ/2 

Vector subtraction 

Suppose there are two vectors A and B, shown in figure A and we have to subtract B and A. It is just the same thing as adding vectors – B to A. The resultant is shown in figure B. 

                                    vector-subtraction 
                                               Figure (A) 


                                        resultant-of-vector-subtraction 
                                              Figure (B) 


           vector-subtract 

Properties of Vector Addition 

    1. Vector addition is commutative

        i.e        
                 example-of-vector-addition 
                                        vector-addition-is-commutative 

     2. Vector addition is associative 

      i.e  
             example-of-associative-vector-addition

Addition and Subtraction of Vectors



                                    vector-addition-is-associative 

Magnitude and direction of a+b

Let angle between vector a and b be θ

In the figure vector (OA) = vector a , vector (AB) = vector b 

From     ADB

                                 AD = b cos θ

                                  BD = b sin θ

                              magnitude-and-direction-of-vectors 
In right angled ODB

                              OD = a + b cosθ

                  BD = b sin θ        Therefore  OB = √(OD2+BD2 )

                      => |a +b |=√(a2+b2+2ab cos θ)

                   |a +b |max   =  a+b   when θ = 2n

                   |a +b |min   =  |a – b|   when θ = (2n + 1)
                                       (where n = 0, 1, 2, …..)

If a + b  is inclined at an angle α with vector a , then

                        tan α = ((b sin θ)/(a+b cosθ))

AskIITians provides free study material for IIT JEE, AIEEE and other engineering entrance examinations. Vectors, their addition and subtraction rules, law of parallelogram addition of vectors are very important from engineering entrance exam point of view as there are many situations in which these concepts and rules are applied. Many a times questions are directly asked from these topics.
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