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Lecture 4 MATH 127 (Maj Sultana)

MATH 127 

Vector Analysis


Triple Product


1.  A.(B×C)  

2.  A×(B×C)  

3.  (A.B).C  


    • Projection

i) Scaler Projection:  ba|a|  

ii) Vector Projection:  a.b|a|2a  

    The scaler projection of  b and  a is the length of the AB (shown in the figure). The vector projection of  b onto  a is the vector with this length at the point A point in the same direction as  a .

    This quantity is also called the component of  b in the  a direction. And, the vector projection is merely the unit vector  a|a times the scalar projection of  b onto  a .

The scalar projection of  b onto  a is proj  ba=b.aa  

and the vector projection of    b onto  a is proj  ba=a.b|a|2a .



    ♦ Direction Cosines: 

  cosα=x|r|=xx2+y2+z2  

  cosβ=y|r|=yx2+y2+z2  

  cosγ=z|r|=zx2+y2+z2  

Theorem:  cos2α+cos2β+cos2γ=1  


    Example: Find direction cosines of a vector represented by  PQ where  P(2,3,5) and Q(1,0,1) are two point.

SolutionPQ=(12)ˆi+(0+3)ˆj+(15)ˆk=ˆi+3ˆj5ˆkNow,α=cos1(112+32+62)=cos1(146) =98.47β=cos1(312+32+62)=cos1346 =63.74θ=cos1(612+32+62)=cos1646 =152.2  


 

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