, I. Vectors
i. I. Vectors and scalars
Scalars :
i. Definition scalar :
A scalar is a number which describes some physical
-
quentieg
um
g Temperature T
e.
Kelvin
.
,
A
,
scalar comes with
it second
Time
-
,
→
a set
of units
Density P
kgm
-
, ,
A scalar be constant also
but
depend
2.
can
, can on another quantity .
THI denote that
temperature T depends time t
g
e. -
on
Vectors :
i. Definition vectors : are quarries with both direction and a ( scalar )
magnitude .
"
e.
g
.
velocity , I , ms
-2
Each vector also
•
Acceleration , a , ms
-
has unit associated
Force I N
'
, , to it
"
-
The force I exerted by an electric field , NC
"
The
force I exerted
by a
magnetic field , T
Notation
U→
2.
usually
:
We underline I arrow
or
place top to
.
an on
mum
mum
denote the u is a vector
•
Bald letter
for print
} .
As with scalars ,
a vector can
depend on some other
gmonties
e.
g E th
Directions talk about directions need frame reference
4. : To we a
of
0
-
Origin
5- position vector :
think of
We can a vector I as
representing the
displacement of a
point A
in
origin space
relative
>
to the
.
.
A
a
I
.
0A.
a- is
position .
vector
of A y
=
,
the
c.
Magnitude
:
6.
•
C
magnitude
= IB71 =
III
g
e.
7
I
8
B
7- Unit Vector :
by £
we call vector of 1 unit vector and denote
length it
we
any a
me
8. A vector can
represent the
displacement between two distinct pairs of
points
, •
A .
C
→ both vectors have the same
7
>
and
°
&
•
direction
magnitude , so
they are the same vector
B
i. II = BE
f. Observations :
In words the direction and under
• other .
magnitude of a vector are invariant
translation .
scalar
magnitude of vector is invariant under
•
The a rotations .
1. 2 . Vector
Algebra
Vector addition and the zero vector :
'
from 0 travel I 0A A
starting the
origin along vector point
=
i. , some to a .
Then travel
along vector to ,
and reach the
point C
A.
±
a-
7
.
(
Y c
a.
-
have
The
point
C must
already a
position vector
→
I = 0C
A
-
.
. I + I =
I
b-
I s
addition
z .
Properties of vector :
+
<
① a- + I = I + I [ vector addition is commutative ] d U
Ta
-
±
② 11th ) + of =
a- + It + d- I [ vector addition is associative ]
( III a)
The
position vector of the
origin ,
0 is
zeirovec-or.fm =
considered don't
It is as a vector .
Only vector have direction
③ a- + A =
It I = I [ the zero vector is the
additive
identity I
otit At = e
If OTI =
I ,
then AT = -
I -
I is called the additive inverse of I
4 a- + 1- E) E) E tbf )
E- I
=
1- + I =
I =
a- +
Scalar
Multiplication →
:
• let a- = 0A and I =
.
Also I =
tea
.
.
.
I has the same direction as I ,
but
only ¥ the
length
i. In
general , for any
real number R . we combine it a vector a to
get a new vector
I =
the
•
when I > a , I has the same direction as I
when I I has the direction as I
20
opposite
- .
e.
g. 7
£
> -
f
Ee L
• when I = a , I = a
Two vectors are
parallel if one is of the form DX other vector
( or anti -
parallel when i ca )
, of scalar multiplication
Properties
:
> .
① I X I = I
② For scalar X
any ,
he + it = A
IITII
③ For scalar and
any X ll .
XI -1
ME = 1 Atm ) I
④ For scalar and
any X µ ,
Rima ) = KM ) I
1. 3 Vector and Coordinate Bases
space
Vector :
space
addtion
1-
Definition Vector space , V :
any hmm
collection of objects equipped with the two
operations 1 Vector
and scalar multiplication ,
satisfying the eight properties outlined I
vectors
z . Vector :
The
objects
um
themself are then
referred to as
space III space )
will restrict attention to the real vector ( two dimensional Euclidean and
3 We our
-
.
.
IRI ( three - dimensional Euclidean
space
) in this course .
.
Scalars are the real number IR
m u m
I A frame of reference 1
origin axis axis and axis
× y Z
together with
:
4 an - -
.
a
-
.
, ,
unit
length I
to ,o a)
• The origin ,
&
PIX y ZI Cartesian Coordinates
-
, '
position vector of P : IF = I =
( If )
of Ipl =] x 't
y -122
'
magnitude
• a vector :
•
vector Addition : If I
=
( £! ) and I =
(If;) , then :
" ± -1%1+11*1=1%11%1
scalar
multiplication for scalar R in IR
•
:
any
NII '
Questions :
suppose that a. =
1¥ ) and b- =/! ) .
Then :
al ath
att -
-
1%1+111=111
b-I -3A
→a =
-4%1 =
II )
4 lol
lol =/ o't 5+1-412 = 5
is .
We can write
any
vector p= I '¥ ) in IR3 in a
unique way
as the linear combination :
(E) =\ ! ) +
yl ! ) +
ZI ! ) = +
yitzk
• The vector I , I , I are called the
¥ivectors
i. I. Vectors and scalars
Scalars :
i. Definition scalar :
A scalar is a number which describes some physical
-
quentieg
um
g Temperature T
e.
Kelvin
.
,
A
,
scalar comes with
it second
Time
-
,
→
a set
of units
Density P
kgm
-
, ,
A scalar be constant also
but
depend
2.
can
, can on another quantity .
THI denote that
temperature T depends time t
g
e. -
on
Vectors :
i. Definition vectors : are quarries with both direction and a ( scalar )
magnitude .
"
e.
g
.
velocity , I , ms
-2
Each vector also
•
Acceleration , a , ms
-
has unit associated
Force I N
'
, , to it
"
-
The force I exerted by an electric field , NC
"
The
force I exerted
by a
magnetic field , T
Notation
U→
2.
usually
:
We underline I arrow
or
place top to
.
an on
mum
mum
denote the u is a vector
•
Bald letter
for print
} .
As with scalars ,
a vector can
depend on some other
gmonties
e.
g E th
Directions talk about directions need frame reference
4. : To we a
of
0
-
Origin
5- position vector :
think of
We can a vector I as
representing the
displacement of a
point A
in
origin space
relative
>
to the
.
.
A
a
I
.
0A.
a- is
position .
vector
of A y
=
,
the
c.
Magnitude
:
6.
•
C
magnitude
= IB71 =
III
g
e.
7
I
8
B
7- Unit Vector :
by £
we call vector of 1 unit vector and denote
length it
we
any a
me
8. A vector can
represent the
displacement between two distinct pairs of
points
, •
A .
C
→ both vectors have the same
7
>
and
°
&
•
direction
magnitude , so
they are the same vector
B
i. II = BE
f. Observations :
In words the direction and under
• other .
magnitude of a vector are invariant
translation .
scalar
magnitude of vector is invariant under
•
The a rotations .
1. 2 . Vector
Algebra
Vector addition and the zero vector :
'
from 0 travel I 0A A
starting the
origin along vector point
=
i. , some to a .
Then travel
along vector to ,
and reach the
point C
A.
±
a-
7
.
(
Y c
a.
-
have
The
point
C must
already a
position vector
→
I = 0C
A
-
.
. I + I =
I
b-
I s
addition
z .
Properties of vector :
+
<
① a- + I = I + I [ vector addition is commutative ] d U
Ta
-
±
② 11th ) + of =
a- + It + d- I [ vector addition is associative ]
( III a)
The
position vector of the
origin ,
0 is
zeirovec-or.fm =
considered don't
It is as a vector .
Only vector have direction
③ a- + A =
It I = I [ the zero vector is the
additive
identity I
otit At = e
If OTI =
I ,
then AT = -
I -
I is called the additive inverse of I
4 a- + 1- E) E) E tbf )
E- I
=
1- + I =
I =
a- +
Scalar
Multiplication →
:
• let a- = 0A and I =
.
Also I =
tea
.
.
.
I has the same direction as I ,
but
only ¥ the
length
i. In
general , for any
real number R . we combine it a vector a to
get a new vector
I =
the
•
when I > a , I has the same direction as I
when I I has the direction as I
20
opposite
- .
e.
g. 7
£
> -
f
Ee L
• when I = a , I = a
Two vectors are
parallel if one is of the form DX other vector
( or anti -
parallel when i ca )
, of scalar multiplication
Properties
:
> .
① I X I = I
② For scalar X
any ,
he + it = A
IITII
③ For scalar and
any X ll .
XI -1
ME = 1 Atm ) I
④ For scalar and
any X µ ,
Rima ) = KM ) I
1. 3 Vector and Coordinate Bases
space
Vector :
space
addtion
1-
Definition Vector space , V :
any hmm
collection of objects equipped with the two
operations 1 Vector
and scalar multiplication ,
satisfying the eight properties outlined I
vectors
z . Vector :
The
objects
um
themself are then
referred to as
space III space )
will restrict attention to the real vector ( two dimensional Euclidean and
3 We our
-
.
.
IRI ( three - dimensional Euclidean
space
) in this course .
.
Scalars are the real number IR
m u m
I A frame of reference 1
origin axis axis and axis
× y Z
together with
:
4 an - -
.
a
-
.
, ,
unit
length I
to ,o a)
• The origin ,
&
PIX y ZI Cartesian Coordinates
-
, '
position vector of P : IF = I =
( If )
of Ipl =] x 't
y -122
'
magnitude
• a vector :
•
vector Addition : If I
=
( £! ) and I =
(If;) , then :
" ± -1%1+11*1=1%11%1
scalar
multiplication for scalar R in IR
•
:
any
NII '
Questions :
suppose that a. =
1¥ ) and b- =/! ) .
Then :
al ath
att -
-
1%1+111=111
b-I -3A
→a =
-4%1 =
II )
4 lol
lol =/ o't 5+1-412 = 5
is .
We can write
any
vector p= I '¥ ) in IR3 in a
unique way
as the linear combination :
(E) =\ ! ) +
yl ! ) +
ZI ! ) = +
yitzk
• The vector I , I , I are called the
¥ivectors
