.9
9
1 1 : Euclidean Spaces
~nation
.
*
R- Set
of
all real numbers
*
N- set
of
all
-
23
-
integers
positive
1 1 1 :
~finition
.
.
of
set
the
elements
all
r-dimensional
on
Euclidean
an
R
n-ordered
with
devoted
and
space
by
coordinates
xu)
3/2
=
viER and
:
....
1
In
it
point
*
iR'-x-axis
*
IR -> My-plane
*
IR3
->
Even
R"-space (IR" for short)
called
is
numbers
in iRh
myz-space
though
Definition
1
:
1
there
is
no
IR"for
to show
geometry
434
but these spaces
,
exist.
2 :
.
-
Let
Plu ,
O
origin
42 ,
and
un)
....,
at
ending
:
·
*
nector
A
has
* The
difference
point
p
(b)
neither
distance
The
directed
The
.
called
is
line
nector
a
between the points
or
and
nectors
the
if
and
.
p
all
are
If
P
is
not the
the
is
origin
origin
called
is
zero
Playe
point
a
at the
segment starting
n-vector , and
D
is
said
is
be
to
,
-
op
then
,
lengtho
the
is
called the
is
called
while
rector
zero
a
op
then
and ends at
un)
direction
a
nor
....,
that
and
but
point ,
a
a
=
has
vector
a
and
nector
length
both
which
,
n)"is
..,
vector
.
non-zero
non-zero
a
(x ,, 42 ,.
vector
zero
the
by
denoted
>
length
from origin
P
point
M"
in
component.
nonzero
between
has
starts
O
one
point
a
the
components
whose
least
at
P
be
vector
that
a
direction ; which
a
the
is
distance
between
.
and P
Definition
1 1 3
.
:
.
- a re
Plu
Let
(y1-x
(1
.
:
Hence ,
if
you
terminal
vector
space
xz
IRh
·
is
defined
ye
-
vector
is
x2
,
equal
...,
yu
points
-
.
R denote the point
at
starting
and
P
ending
at
$
xn)
in
anywhere
a
to
Rh Let
in
segment
by
a
rector
non-zero
a
,,
be
directed line
the
(y1 -x
=
yn)
..,
the
the
same
my-plane
vector
without
may
its
changing
initial and
have
points.
, ...,
a
R
resulting
this establishes
,
and
=
more
$(y11y21
and
yu-xn)
...,
·9
direction , the
P(x
1
un)
vector
a
2)
.
...,
,
42-42
,,
called
is
uz
,
=
dr)
(n ,
,
2
one-to-one
a
Rh
in
,
defined
...,
un)
in
,
there
,
1 1 1
:
there
.
can
1R" that
points and vectors in
unique nector op corresponding to the point
relation between
is
a
unique
is
a
be
treated
point (M
as the set
,
of
Mc
all
...,
Mu)
corresponding
vectors
in
.
uph
to the
is
p
.
,
for
each point
Conversely for
nector a
,
.
Hence
,
the