Single Math B part 1(first-year Math)

, 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