13.2-4 Properties of Vector Operations

13.2-4 Properties of Vector Operations - Properties of...

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Properties of Vector Operations Addition and Scalar Multiplication 1 . ~a + ~ b = ~ b + ~a 2 . ~a + ( ~ b + ~ c ) = ( ~a + ~ b ) + ~ c 3 . ~a + ~ 0 = ~a 4 . ~a + ( - ~a ) = ~ 0 5 . c ( ~a + ~ b ) = c~a + c ~ b 6 . ( c + d ) ~a = c~a + d~a 7 . ( cd ) ~a = c ( d~a ) 8 . 1 ~a = ~a Dot Product The dot product is defined by ~a = ( a 1 , a 2 , a 3 ) , ~ b = ( b 1 , b 2 , b 3 ) = ~a · ~ b = a 1 b 1 + a 2 b 2 + a 3 b 3 and obeys 0 . ~a, ~ b are vectors and ~a · ~ b is a number 1 . ~a · ~a = k ~a k 2 2 . ~a · ~ b = ~ b · ~a 3 . ~a · ( ~ b + ~ c ) = ~a · ~ b + ~a · ~ c 4 . ( c~a ) · ~ b = c ( ~a · ~ b ) 5 . ~ 0 · ~a = 0 6 . ~a · ~ b = k ~a k k ~ b k cos θ 7 . ~a · ~ b = 0 ⇐⇒ ~a = ~ 0 or ~ b = ~ 0 or ~a ~ b In property 6, θ is the angle between ~a and ~ b . 1
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Cross Product The cross product is defined by ~a = ( a 1 , a 2 , a 3 ) , ~ b = ( b 1 , b 2 , b 3 ) = ~a × ~ b = ( a 2 b 3 - a 3 b 2 , a 3 b 1 - a 1 b 3 , a 1 b 2 - a 2 b 1 ) and obeys 0 . ~a, ~ b and ~a × ~ b are all vectors in three dimensions 1 . ~a × ~ b ~a, ~ b 2 . k ~a × ~ b k = k ~a k k ~ b k sin θ 3 . ˆ ı × ˆ = ˆ k, ˆ × ˆ k = ˆ ı, ˆ k × ˆ ı = ˆ 4 . ~a × ~ b = k ~a k k ~ b k sin θ ˆ n 5 . ~a × ~ b = 0 ⇐⇒ ~a = ~ 0 or ~ b = ~ 0 or ~a k ~ b 6 . ~a × ~ b = - ~ b × ~a 7 . ( c~a ) × ~ b = ~a × ( c ~ b ) = c ( ~a × ~ b ) 8 . ~a × ( ~ b + ~ c ) = ~a × ~ b + ~a × ~ c 9 . ~a · ( ~ b × ~ c ) = ( ~a × ~ b ) · ~ c 10 . ~a × ( ~ b × ~ c ) = ( ~ c · ~a ) ~ b - ( ~ b · ~a ) ~ c In properties 2 and 4,
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