1.  A ABAB 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.

2. associative law for addition if a,b,c  V then a  ( b  c ) = ( a  b)  c example 1: A B C BCBC ABAB 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 ]

3. 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.

4. 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

5. 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 ABAB A B ABAB BABA if a, b  V then a  b = b  a example 2

6. 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

7. 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 1A1A example 2

8.  (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)

9. (   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

10. 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.

11. 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

12. 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)

13. 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

14. 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

15. 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

16. the scalar product in R n example: [1,-1,2] ○ [2,3,0] = 1·2 + (-1)·3 + 2·0 =

17. scalar product of vector quantities For physical vector quantities, we define scalar product through the scalar product of the oriented segments representing them.

18. the magnitude The magnitude of a vector is a number defined by the scalar product: example: magnitude of an oriented segmentA a

19. 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.

20. 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 

21. 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

22. 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.

23. 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