AAB

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 Rn

A = [A1, A2, A3]

B = [B1, B2, B3]

AB = [A1+B1, A2+ B2, A3+ B3]

A = [A1, A2, A3]

are called vectors, if and only if, all eight of the following conditions are satisfied.

associative law for addition

if a,b,c V then a ( b c ) = ( a b) c

example 1:

A

B

C

BCAB

A(BC)A(BC)(AB)C

example 2:

[A1,A2,A3]([B1,B2,B3] [C1,C2,C3]) =

= [A1,A2,A3][(B1+ C1),(B2+ C2),(B3+ C3)] =

= [A1+(B1+ C1), A2+(B2+ C2), A3+(B3+ C3)] =

= [(A1+B1)+ C1, (A2+B2)+ C2, (A3+B3)+ C3] =

= [(A1+B1), (A2+B2), (A3+B3)] [C1,C2,C3] =

= ([A1,A2,A3][B1,B2,B3]) [C1,C2,C3]

additive identity

[A1,A2,A3] [0,0,0] =

= [(A1+0), (A2+0), (A3+0)] =

= [A1,A2,A3]

example 1 example 2

There is such an element 0 V that for each a V, a 0 = a.

additive inverse

[A1,A2,A3] [-A1,-A2,-A3] =

= [A1+(-A1), A2+(-A2), A3+(- A3)] =

= [0,0,0]

For each aV there is (-a) V that a (-a)=0

example 1

example 2

A-A

0

commutative law of addition

[A1,A2,A3][B1,B2,B3]=

= [(A1+B1), (A2+B2), (A3+B3)] =

= [(B1+A1), (B2+A2), (B3+A3)] =

= [B1,B2,B3] [A1,A2,A3]

example 1

AB

A

BAB

BA

if a, b V then a b = b a

example 2

associative law for multiplication

([A1,A2,A3]) =

= [(A1), (A2), (A3)]=

= [(A1), (A2), (A3)]=

=[()A1, ()A2, ()A3)]=

=() [A1,A2,A3]

If R and a V then ( a ) = () a

example 1

AA

(A)(A)

()A)

example 2

multiplicative identity

1 [A1,A2,A3] =

= [1A1,1A2,1A3] =

= [A1,A2,A3]

For every a V, 1 a = a

example 1

A

1A

example 2

(AB)(AB)

distributive law

([A1,A2,A3][B1,B2,B3]) =

= [(A1+B1), (A2+B2), (A3+B3)] =

= [(A1+B1), (A2+B2), (A3+B3)] =

= [A1+B1, A2+B2, A3+B3] =

= ([A1, A2, A3][B1, B2, B3])=

= [A1,A2,A3] [B1,B2,B3]

if R, a,b V then (a b) = ( a) ( b)

example 1

A

B

( A)( B)example 2

( A)

( B)

( a) ( a)( a) ( a)

distributive law

(+)[A1,A2,A3] =

= [(+)A1,(+)A2,(+)A3] =

= [(A1+A1),(A2+A2),(A3+A3)]=

= [A1,A2,A3] [A1,A2,A3] =

= [A1,A2,A3] [A1,A2,A3]

if ,R, aV then (+) a = ( a) ( a)

example 1

A

A A

(+) a

example 2

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.

the base The smallest sets of vectors {e1,… en}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)

n

1iiix ex

The dimension of the space is the number of the elements in the base.

scalar component

vector component

isomorphism

Vector spaces of the same dimension are isomorphic, which means that there is a one-to-one function F: V1V2, 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:

baba FFF 222111

a

b

1a 1 1b F(a)

F(b)

2F(a) 2 2 F(b)

A = [ , , ]

oriented segment triad of numbers(Cartesian system)

A

i j

k

x

y

z

Ax = Ax i

Ay = Ay j

Az = Az k

A = (Ax i) (Ay j) (Az k )

Ax Ay Az

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

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

cos abBA

example: scalar product of perpendicular segments of unit length

090cos11 BA

the scalar product in Rn

][ ,...a,a 21A

][ ,...b,b 21B

n

1iiibaBA

example:

[1,-1,2] ○ [2,3,0] = 1·2 + (-1)·3 + 2·0 = -1

scalar product of vector quantities

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

the magnitude

The magnitude of a vector is a number defined by the scalar product:

2aaaa

example: magnitude of an oriented segment A

aaa 0cosA 22AA

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.

ba

kjikjiba

332211

332313

322212

312111

321321

bababa

0cosba90cosba90cosba

90cosba0cosba90cosba

90cosba90cosba0cosba

ˆbˆbˆbˆaˆaˆa

angle between vectors

cos 1

a b

a b

The angle between two vectors is defined by the scalar product

(The angle defined above coincides with the angle between the oriented segments.)

A

B

example:Find the angle between [2,0] and [1,1].

cos 12 2 2 2

2 1 0 1

2 0 1 145

i

j

]0,2[A

]1,1[B

x

y

= 45

projection of a vector

For any arbitrary vector and a unit vector , vector A ie

iii eeAA ˆ)ˆ(

is called the projection of vector in the direction of vector .

A

ie

A

ix

Ax

Ax = ( a ·1· cos ) • i

Ax = ( a cos ) example a

Ax

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.

n

1iˆˆ ii eeAA

n

1i

n

1ii ˆA ii AeA

the components

example: 2D space

A

x

y

AxAx

Ay

Ay

Ax = A ○ i = = A 1 cos = A cos Ax = A cos i

Ay = A cos = A sin

Ay = A sin j