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Lecture 2 [06 Feb, 2020] MATH 127 (Maj Sultana)

 MATH 127 

Vector Analysis





    ♦ Vector addition: The sum of two vector A and B is a vector  C , which obtained by placing the initial point of  B on the final point of  A and the drawing a line from the initial point of  A to the final point of  B .

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    ♦ Scaler product: The product of a scaler say m times a vector  A is another vector  B , where  B has the direction as  A but the magnitude is changed that is  |B|=m|A|  

    • Properties: 

1. Commutative law for addition:  A+B=B+A  

2. Associative law for addition:  A+(B+C)=(A+B)+C  

3. Commutative law for multiplication:  mA=Am  

4. Associative law for multiplication:  (m+n)A=mA+nA , where m and n are two different scalers.

5. Distributive law:  m(A+B)=mA+mB  




    Vector Multiplication: There are two types of multiplication.

1. Scaler or Dot product          2. Vector or Cross product


    Dot Product: 

A geometric representation of dot product: The scaler or dot product of two vectors,  A   and  B denoted by  A.B , is defined as the product of the magnitude of vector times the cosine of the angle between them.

  A.B=|A| |B|cosθ        [θ=mnimum angle]  


A cartesian representation of dot product:  textrmIA=A1ˆi+A2ˆj+A3ˆk and B=B1ˆi+B2ˆj+B3ˆk then AB=A1B1+A2B2+A3B3  


    • Properties of Dot product:

1.  A.B=B.A  

2.  A.(B+C)=A.B+A.C  

3.  m(A.B)=(mA.B)=A.(mB)=(A.B)m


    Some Important results:

1.  ˆi.ˆi=|ˆi||ˆi|cos0=1 ˆj.ˆj=1 ˆk.uvjk=1  

2.  ˆi.ˆj=|ˆi|.|ˆj|cos90=0 ˆj.ˆk=0 ˆk.ˆi=0  

3. If we have  A.B=0 means, neither the magnitude of  A nor  B is 0, then  A and  b must be perpendicular.

  A.B=0and|A|0 and |B|0           then, θ=90  



    Example: What is the angle between of two vectors  A=2ˆi+2ˆj and B=4ˆi3ˆj ?

Solution:  A.B=126=6|A|=22+22=22|B|=62+(3)2=45   θ=cos1(6290)          =71.56




    ♦ Cross product: 

    A geometric representation: The cross product of two vector  A and  B denoted by  A×B is defined as the product of the magnitudes of the vectors times the sine of the angle between them with a unit vector  ˆn .

  A×B=|A|.|B|sinθ ˆn is the unit vector at right angles to both A and A


    A cartesian representation: 

 If A=A1ˆi+A2ˆj+A3ˆkB=B1ˆi+B2ˆj+B3ˆk   

Then  A×B=|ˆiˆjˆkA1A2A3B1B2B3|=ˆi(A2B3B2A3)+ˆj(A3B1A1B3)+ˆk(A1B2A2B1)  



    Example: If  A=2ˆi+ˆjˆk and B=3ˆi+4ˆj+ˆk find  A×B                        


Solution: 

  A×B=|ˆiˆjˆk211341|=6ˆi+ˆj+11ˆk  



    • Properties of Cross product: 

1. If  A×B=0 then  A and  B will be parallel vector.

2.  A×B0 provided that  A×B is orthogonal to both  A and  B  

3. If  u,v and w vectors and c is a number then

  i) u×v=(v×u)ii) u×(v+w)=u×v+u×wiii) (cu)×v=uvu×(cv)=c(u×v)iv) u(v×w)=(u×v)wv) u(v×w)=|u1u2u3v1v2v3w1w2w3|  



  



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