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Lecture 3 [11 Feb, 2020] MATH 127 (Maj Sultana)

 MATH 127

Vector Analysis




    • Find the angle between two vectors where  A=2ˆi+2ˆj and B=6ˆi3ˆj  

  θ=cos1A.B|A|.|B|=cos112622.45=71.56  


    • Important properties: 

1. The area of the parallelogram is given,  A =  |A×B|  

2. The volume of the parallelepiped is given, V =    |A.(B×C|)  

3. Are of Triangles =  12|A×B|  

4. If the three vectors a co-planar then  (A×B).C=0 or A(B×C=0)  

5.  (A×B).C=0   implies two of the vectors parallel.

6.  (A×B).C=A.(B×C)=B.(C×A)   = same volume.



    • Position vector:


  OP =xˆi+yˆj+zˆkPQ =QP=(x1x)ˆi+(y1y)ˆj+(z1z)ˆkOQ =x1ˆi+y1ˆj+z1ˆk  


    Example: A plane is defined by any three points that are in the plane. If a plane contains the point  P=(1,0,0),Q(1,1,1) and R(2,1,3)   then find a vector that is orthogonal to the plane.

 Solution: PQ=ˆj+ˆkPR=ˆiˆj+3ˆk|PQ×PR=|ˆiˆjˆk011113|  



    Example: Find the area of the parallelogram with vectors  P(2,2,0),Q(9,2,0),R(10,3,0) and S(3,3,0)  

Solution:    QP=7ˆiQR=ˆi+ˆj|QP×QR=|ˆiˆjˆk700110|=|7ˆk|=7sqr unit  



    Example: Find the area of the triangle with vectors  A(1,0,0),B(0,1,0),C(0,0,1) .

SolutionABC=12|AB×AC=12|ˆiˆjˆk110101|=12|ˆi+ˆj+ˆk|=1212+12+12=32 sqr unit  



    Example: find the volume of the parallelepiped, determined by the vectors  (1,1,1),(4,7,2),(3,2,1), and (5,4,3)  



    Example: Determine, whether the three vectors  A=ˆi+4ˆj7ˆk,B=2ˆiˆj+4ˆk, and C=9ˆj+18ˆk   lie in the same plane or not.

SolutionC(A×B)=|0918147214|=9(4+14)+18(18)=0

these vectors exists in the same plane.

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