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A ABAB VECTORS Elements of a set V for which two operations are defined: internal (addition) and external (multiplication by a number), example 1: oriented segments A B example 2: ordered sets of numbers R n 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 ] A = [ A 1, A 2, A 3 ] are called vectors, if and only if, all eight of the following conditions are satisfied.
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associative law for addition if a,b,c V then a ( b c ) = ( a b) c example 1: A B C BCBC ABAB A (B C) (A B) C example 2: [A 1,A 2,A 3 ] ([B 1,B 2,B 3 ] [C 1,C 2,C 3 ]) = = [A 1,A 2,A 3 ] [(B 1 + C 1 ),(B 2 + C 2 ),(B 3 + C 3 )] = = [A 1 +(B 1 + C 1 ), A 2 +(B 2 + C 2 ), A 3 +(B 3 + C 3 )] = = [(A 1 +B 1 )+ C 1, (A 2 +B 2 )+ C 2, (A 3 +B 3 )+ C 3 ] = = [(A 1 +B 1 ), (A 2 +B 2 ), (A 3 +B 3 )] [C 1,C 2,C 3 ] = = ([A 1,A 2,A 3 ] [B 1,B 2,B 3 ]) [C 1,C 2,C 3 ]
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additive identity [A 1,A 2,A 3 ] [0,0,0] = = [(A 1 +0), (A 2 +0), (A 3 +0)] = = [A 1,A 2,A 3 ] example 1example 2 There is such an element 0 V that for each a V, a 0 = a.
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additive inverse [A 1,A 2,A 3 ] [-A 1,-A 2,-A 3 ] = = [A 1 +(-A 1 ), A 2 +(-A 2 ), A 3 +(- A 3 )] = = [0,0,0] For each a V there is (-a) V that a (-a)=0 example 1 example 2 A -A 0
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commutative law of addition [A 1,A 2,A 3 ] [B 1,B 2,B 3 ]= = [(A 1 +B 1 ), (A 2 +B 2 ), (A 3 +B 3 )] = = [(B 1 +A 1 ), (B 2 +A 2 ), (B 3 +A 3 )] = = [B 1,B 2,B 3 ] [A 1,A 2,A 3 ] example 1 ABAB A B ABAB BABA if a, b V then a b = b a example 2
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associative law for multiplication ( [A 1,A 2,A 3 ]) = = [( A 1 ), ( A 2 ), ( A 3 )]= = [ ( A 1 ), ( A 2 ), ( A 3 )]= =[( )A 1, ( )A 2, ( )A 3 )]= =( ) [A 1,A 2,A 3 ] If R and a V then ( a ) = ( ) a example 1 A A ( A) ( ) A) example 2
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multiplicative identity 1 [A 1,A 2,A 3 ] = = [1A 1,1A 2,1A 3 ] = = [A 1,A 2,A 3 ] For every a V, 1 a = a example 1 A 1A1A example 2
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(A B) distributive law ([A 1,A 2,A 3 ] [B 1,B 2,B 3 ]) = = [(A 1 +B 1 ), (A 2 +B 2 ), (A 3 +B 3 )] = = [ (A 1 +B 1 ), (A 2 +B 2 ), (A 3 +B 3 )] = = [ A 1 + B 1, A 2 + B 2, A 3 + B 3 ] = = ([ A 1, A 2, A 3 ] [ B 1, B 2, B 3 ])= = [A 1,A 2,A 3 ] [B 1,B 2,B 3 ] if R, a,b V then (a b) = ( a) ( b) example 1 A B ( A)( B)( A)( B) example 2 ( A)( A) ( B)( B)
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( a) ( a) distributive law ( + ) [A 1,A 2,A 3 ] = = [( + )A 1,( + )A 2,( + )A 3 ] = = [( A 1 + A 1 ),( A 2 + A 2 ),( A 3 + A 3 )]= = [ A 1, A 2, A 3 ] [ A 1, A 2, A 3 ] = = [A 1,A 2,A 3 ] [A 1,A 2,A 3 ] if , R, a V then ( + ) a = ( a) ( a) example 1 A A A A A (+) a(+) a example 2
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Vector quantities A quantity that obeys the same rules of combination as vectors is a vector quantity. Each vector quantity can be represented isomorphically by a vector, but cannot be represented by a number.
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the base The smallest sets of vectors {e 1,… e n } V is called the base of the vector space, if and only if each vector x can be represented as (linear combination of the base vectors) The dimension of the space is the number of the elements in the base. scalar component vector component
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isomorphism Vector spaces of the same dimension are isomorphic, which means that there is a one-to-one function F: V 1 V 2, that allows us to predict the result of a combination of vectors in one vector space by combining appropriate vectors in the other vector space: a b 1 a 1 1 b F(a) F(b) 2 F(a) 2 2 F(b)
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A = [,, ] oriented segment triad of numbers (Cartesian system) A i j k x y z A x = A x i A y = A y j A z = A z k A = (A x i) (A y j) (A z k ) AxAx AyAy AzAz
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the scalar product a ○ b = b ○ a (commutative) ( a) ○ b = (a ○ b)(associative) (a b) ○ c = (a ○ c) + (b ○ c) (distributive) a ○ a 0; a ○ a = 0 a = 0
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the scalar product of oriented segments where a and b are the lengths of the segments and is the angle between the segments A B a b example: scalar product of perpendicular segments of unit length
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the scalar product in R n example: [1,-1,2] ○ [2,3,0] = 1·2 + (-1)·3 + 2·0 =
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scalar product of vector quantities For physical vector quantities, we define scalar product through the scalar product of the oriented segments representing them.
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the magnitude The magnitude of a vector is a number defined by the scalar product: example: magnitude of an oriented segmentA a
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theorem The scalar product of two oriented segments is equal to the scalar product of the corresponding triads (vectors of scalar components) in a Cartesian system.
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angle between vectors The angle between two vectors is defined by the scalar product (The angle defined above coincides with the angle between the oriented segments.) example: Find the angle between [2,0] and [1,1]. x y = 45
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projection of a vector For any arbitrary vector and a unit vector, vector is called the projection of vector in the direction of vector. A i x AxAx AxAx = ( a ·1· cos ) i A x = ( a cos ) example a AxAx
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theorem The sum of the vector projections of a vector in all mutually perpendicular (in the sense of the scalar product) directions is equal to the vector. The projections constitute the vector components of the vector.
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the components example: 2D space A x y AxAx AxAx AyAy AyAy A x = A ○ i = = A 1 cos = A cos A x = A cos i i A y = A cos = A sin A y = A sin j j
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