Advanced Concepts in Vector Calculus: Properties, Problems, and Solutions
Discover the intricacies of scalar triple products, their properties, and applications through solved problems. Understand coplanarity and cyclic permutations in three-dimensional vectors.
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DEPARTMENT OF MATHEMATICS SGGSJ GOVERNMENT COLLEGE PAONTA SAHIB
COURSE NAME: VECTOR CALCULUS COURSE CODE : MATH310TH
CONTENT SCALER TRIPLE PRODUCT VECTOR TRIPLE PRODUCT
SCALER TRIPLE PRODUCT : If ? , ? , ? are any three vectors, then the scalar ( ? ? ) . ? is called the scalar triple product of ? , ? and ? and is denoted by [ ? , ? , ? ] or [ ? ? ? ] . [ ? , ? , ? ] or [ ? ? ? ] = ( ? ? ) . ?
PROPERTIES OF SCALER TRIPLE PRODUCT : PROPERTY 1 : If ? , ? , ? are cyclically permuted, then the value of scalar triple product remains same . i.e. [? ,? ,?] = [? ,? ,?] = [? ,? ,?] . PROPERTY 2 : In scalar triple product the position of dot and cross can be interchanged provided that the cyclic order of the vectors remains same . i.e. ( ? ? ) . ? = ? . ( ? ? ) PROPERTY 3 : The change of cyclic order of vectors in scalar triple product changes the sign of the scalar triple product but not the magnitude. i.e. . [? ,? ,?] = - [? ,? ,?] = - [? ,? ,?] = - [? ,? ,?]
PROPERTY 4 : The scalar triple product of three vectors is zero if any two of them are equal . PROPERTY 5 : For any three vectors ? , ? , ? and scalar then , [ ? ,? ,?] = [? ,? ,?] PROPERTY 6 : : For any three vectors ? , ? , ? and any three scalar l , m , n, then [l l ? , m? , n?] = l m n l m n [? ,? ,?] PROPERTY 7 : The scalar triple product of three vectors is zero if any two of them are parallel or collinear . PROPERTY 8 : The necessary and sufficient condition for three non zero , non- collinear vectors ? , ? , ? to be coplanar is that [? ,? ,?] = 0
PROBLEM 1 : Find [? ,? ,?] , if ? = ? - 2 ? + 3? , ? = 2 ? - 3 ? + ? , ?= 3 ? + ? - 2 ? . SOLUTION : Here, ? = ? - 2 ? + 3 ? ? = 2 ? - 3 ? + ? and ?= 3 ? + ? - 2 ? ? ? ? ? ? ? ? ? [ ? , ? , ? ] = ? = 1 ( 6 1 ) + 2 ( -4-3 ) + 3 ( 2 + 9 ) = 5 14 + 33 = 24.
PROBLEM 2 : If ? = ? + 2 ? + ? , ? = 3? + 2 ? - 7 ? , ?= 5 ? + 6 ? - 5 ? . Show that ? , ? , ? are coplanar. SOLUTION : Here, ? = ? + 2 ? + ? ? = 3 ? + 2 ? - 7 ? ?= 5 ? + 6 ? - 5 ? ? ? ? ? ? ? ? [ ? , ? , ? ] = ? ?
SOLUTION CONTINUES = 1 ( -10 + 42 ) -2 ( -15 + 35 ) + 1 ( 18- 10 ) = 32 40 + 8 = 40 40 = 0. [ ? , ? , ? ] = 0 ? , ? , ? are coplanar.
VECTOR TRIPLE PRODUCT : If ? , ? , ? are any three vectors, then the cross product of ? and ? ? or cross product of ? ? and ? is called the Vector triple product of ? , ? and ? and is written as ? ( ? ? ) or ( ? ? ) ? respectively. EXPANSION FORMULA : # ? ( ? ? ) = ( ? . ? ) ? - ( ? . ? ) ? . # ( ? ? ) ? = ( ? . ? ) ? - ( ? . ? ) ? .
PROBLEM 3 : Prove that : ? { ? ( ? ? ) } = ( ? . ? ) ( ? ? ) . SOLUTION : L.H.S = ? {( ? . ? ) ? - ( ? . ? ) ? } = ( ? . ? ) ( ? ? ) - ( ? . ? ) ( ? ? ) = ( ? . ? ) ( ? ) + ( ? . ? ) ( ? ? ) = ( ? . ? ) ( ? ? ) = R.H.S HENCE PROVED.
PROBLEM 4: Find ( ? ? ) ? , if ? = ? - 2 ? + 3? , ? = 2 ? + ? + ? , ?= ? + ? + 2 ? . SOLUTION : here, ? = ? - 2 ? + 3 ? ? = 2 ? + ? + ? and ?= ? + ? + 2 ? . ? ? ? ? ? ? ? ? ? ? ? = = ( -2 3 ) ? - ( 1 6 ) ? + ( 1 + 4 ) ? = -5 ? +5 ? + 5 ?.
SOLUTION CONTINUES Now, ? ? ? ? ? ? ? ( ? ? ) ? = ? ? = ( 10 - 5 ) ? - ( -10 5 ) ? + ( -5 -5 ) ? = 5 ? + 15 ? - 10 ? .
