Download VECTORS

VECTORS
example 1:
oriented segments
AB
A
B
A
example 2:
ordered sets of numbers Rn
A = [A1, A2, A3]
B = [B1, B2, B3]
AB = [A1+B1, A2+ B2, A3+ B3]
A = [A1, A2, A3]
Elements of a set V for which two operations are defined:
internal  (addition) and external  (multiplication by a
number), 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 2:
1:
(AB)C
[A1,A2,A3]([B1,B2,B3]
[C1,C2,C3]) =
A(BC)
= [A1,A2,A3][(B
BC3+ C3)] =
AB 1+ C1),(B2+ C2),(B
B 2+ C2), A3+(B3+ C3)] =
= [A1+(B1+ C1), A2+(B
= [(A1+B1)+ C1, (A2+B2)+ C2, (A3+B3)+ C3] =
C
= [(A1+B1), (A2+B
),
(A
+B
)]

[C
2
3
3
1,C2,C3] =
A
= ([A1,A2,A3][B1,B2,B3]) [C1,C2,C3]
additive identity
There is such an element 0 V that for each a V, a  0 = a.
example 1
example 2
[A1,A2,A3] [0,0,0] =
= [(A1+0), (A2+0), (A3+0)] =
= [A1,A2,A3]
additive inverse
For each aV there is (-a) V that a  (-a)=0
example 1
A
0
-A
example 2
[A1,A2,A3] [-A1,-A2,-A3] =
= [A1+(-A1), A2+(-A2), A3+(- A3)] =
= [0,0,0]
commutative law of addition
if a, b V then a  b = b  a
example 2
example 1
BA
AB
B
A
[A1,A2,A3][B1,B2,B3]=
= [(A1+B1), (A2+B2), (A3+B3)] =
= [(B1+A1), (B2+A2), (B3+A3)] =
= [B1,B2,B3]  [A1,A2,A3]
associative law for multiplication
If   R and a V then   (   a ) = ()  a
example 1
example 2
A
A
(A)
()A)
([A1,A2,A3]) =
= [(A1), (A2), (A3)]=
= [(A1), (A2), (A3)]=
=[()A1, ()A2, ()A3)]=
=() [A1,A2,A3]
multiplicative identity
For every a V, 1  a = a
example 2
example 1
A
1A
1  [A1,A2,A3] =
= [1A1,1A2,1A3] =
= [A1,A2,A3]
distributive law
if R, a,b V then   (a  b) = (  a)  (  b)
example 1
(  B)
example 2
(  A)(  B)
(AB)
B
A
(  A)
([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]
distributive law
if ,R, aV then (+)  a = (  a)  (  a)
example 1
(+)  a
(  a)  (  a)
example 2
A
A
A
(+)[A1,A2,A3] =
= [(+)A1,(+)A2,(+)A3] =
= [(A1+A1),(A2+A2),(A3+A3)]=
= [A1,A2,A3]  [A1,A2,A3] =
= [A1,A2,A3]  [A1,A2,A3]
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)
vector component
n
x   xi  ei
i 1
scalar component
The dimension of the space is the number of the
elements in the base.
isomorphism
Vector spaces of the same dimension are isomorphic, which
means that there is a one-to-one function F: V1V2, 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:
F 1 a 1  1 b   2 Fa 2  2 Fb
b
F(b)
2F(a) 2 2 F(b)
a
1a 1 1b
F(a)
oriented segment  triad of numbers
(Cartesian system)
z
A = [ Ax , Ay , Az]
Az = Az k
A = (Ax  i)  (Ay  j)  (Az  k )
k
Ay = Ay j
i
x
Ax = Ax i
j
y
the scalar product
•
•
•
•
a ○b = b ○a
(  a) ○ b =   (a ○ b)
(a  b) ○ c = (a ○ c) + (b ○ c)
a ○ a  0; a ○ a = 0  a = 0
(commutative)
(associative)
(distributive)
the scalar product of oriented segments
 
A  B  ab cos 
b
B

where a and b are the lengths of
the segments and  is the angle
between the segments
A
a
example:
scalar product of perpendicular segments of unit length
 
A  B  11 cos 90  0
the scalar product in Rn
A  [a1 ,a2 ,...]
B  [b1 ,b2 ,...]
n
A  B   aibi
i 1
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:

 
2
a  aa  a
example: magnitude of an oriented segment

2
A  A  A  a 2  cos 0  a
A
a
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.



 
a  b  a1ˆi  a 2ˆj  a 3kˆ  b1ˆi  b 2ˆj  b3kˆ 
 a1b1 cos 0  a1b 2 cos 90  a1b3 cos 90 
 a 2 b1 cos 90  a 2 b 2 cos 0  a 2 b3 cos 90 
 a 3b1 cos 90  a 3b 2 cos 90  a 3b3 cos 0 
 a1b1  a 2 b 2  a 3b3  a  b
angle between vectors
The angle between two vectors is defined by the scalar
product
  cos1
 
ab
 
ab
y

B
(The angle defined above coincides with the
ĵ
angle between the oriented segments.)
 = 45
î
example:
Find the angle between [2,0] and [1,1].
  cos 1
2 1  0 1
 45
2
2
2
2
2  0  1 1
A  [2,0]
B  [1,1]

A
x
projection of a vector

For any arbitrary vector A and a unit vector êi , vector


Ai  (A  eˆ i )  eˆ i

is called the projection of vector A in the direction of
vector êi .
example
A
a
Ax = ( a cos )
Ax = ( a ·1· cos ) • i

x
i
Ax 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.
 n 
A   A  eˆ i   eˆ i
i 1
The projections constitute the vector components of the vector.
 n
n 
A   Ai  eˆ i   Ai
i 1
i 1
the components
example: 2D space
Ax = A ○ i =
= A  1  cos 
= A cos 
y
A
Ay
Ay
Ax = A cos   i

Ay = A cos  = A sin 

Ax
Ax
x
Ay = A sin   j