The Geometry of Euclidean space
1 1
.
:
Vectors in ID & 3D space :
Z
Ya M
az ·
Pla ,, az)
03
· P(U , Uz As) ,
X >
Y
a az
plane
a space
Notation :
IR :
number line
IR2 :
Points in a plane
IR3 :
Points in space
Vector Addition & Scalar Multiplication
:
&
&
az + B2
-
i" 1 Sum :
(a , + bi ,
dz + bz , Az + bs)
air
A
&
2. 0 :
10 ,
0 , 07
a + b,
3 . (-9 ,, -82 -93) ,
:
additive inverse ofA
4 . X =
(x9 ,, && 2 , X9s)
Properties :
1 .
(B)(9 1, 92 93) ,
=
x(BCA , &2 ,
as]] associativity
2. (x + B) (9 ,, dz As),
=
X (9 , &2 , As) + B(a , &z as) , distributivity
3 x10 ,
0 0) =
10 , 0,0
,
.
properties of o
4 .
01a , da 93) ,
=
10 0 8)
. .
Geometry :
a =
(9 ,, Cz 93) ,
=
(9 ,, 0 , 0) + 10 , 92 , 0) + 10 ,
0 ,
as)
As
As
a , (1 8)
0
:
, + +
,
-
i
d,i
= + azj + ask
Q(X : Y'z' pa =
n + o
P = -
jp + o
010 , 0 , 0
(X , Y , z)
·
-Op
=
(xi y' z') , ,
-
(x y z)
, ,
:
(x' -
X , y' y z' z)
-
,
-
, Equations of Lines :
Q l
i =
% :
p + t
a
i,
a ,
O
=
(a , b ,
c) + +(V ,, Vz Vs) ,
=
(a ++ V
, b+ +Vz , C+ +Vs)
X =
a+ +V,
y =
b + +V
z =
C + tV
ei a (5 j)
a
=
+ + -
P
in = a + +5 -
th
·
O
=
(l -t)a + +
EX finding the eg of a line a direction :
: .
in
ala an as) , , ,
i lorg , in
i =
(9 ,, 92 93) ,
+ ti
CX finding the eg
. of a line between 2 points
:
:
ala ,
az 93) , b(b bribs) ,
v =
5 -
a =
(b , -
a, bz Az
-
, by As)
-
i =
a + t
i =
(9 ,, Az ,
as) + + (b , -
a ,, by 02 , by az)
-
1 2
.
:
The inner product ,
length , a distance
inner product
:
& 5
.
=
a ,b, +
azbz + Asks (scalar)
Properties :
I . . , 0
a c .
=
0 when a j =
x 5 a
-
2 .
. =
x( 5) - =
xb
.
3 & (5 + 5) .
= .
5 + a.
H & 5 .
=
5.
.5 =
11 11 III COSIO)
Proof :
115-all =
11 + 115112 Ill @IllI511 cos(t) -
15-2) (5 - a) .
=
lat + 11b(R-IIIIIIIIICOS(E)
5 5
. -
25 +. =
11 + 115112 -
ZII/IIE11cOSIO)
1 1
.
:
Vectors in ID & 3D space :
Z
Ya M
az ·
Pla ,, az)
03
· P(U , Uz As) ,
X >
Y
a az
plane
a space
Notation :
IR :
number line
IR2 :
Points in a plane
IR3 :
Points in space
Vector Addition & Scalar Multiplication
:
&
&
az + B2
-
i" 1 Sum :
(a , + bi ,
dz + bz , Az + bs)
air
A
&
2. 0 :
10 ,
0 , 07
a + b,
3 . (-9 ,, -82 -93) ,
:
additive inverse ofA
4 . X =
(x9 ,, && 2 , X9s)
Properties :
1 .
(B)(9 1, 92 93) ,
=
x(BCA , &2 ,
as]] associativity
2. (x + B) (9 ,, dz As),
=
X (9 , &2 , As) + B(a , &z as) , distributivity
3 x10 ,
0 0) =
10 , 0,0
,
.
properties of o
4 .
01a , da 93) ,
=
10 0 8)
. .
Geometry :
a =
(9 ,, Cz 93) ,
=
(9 ,, 0 , 0) + 10 , 92 , 0) + 10 ,
0 ,
as)
As
As
a , (1 8)
0
:
, + +
,
-
i
d,i
= + azj + ask
Q(X : Y'z' pa =
n + o
P = -
jp + o
010 , 0 , 0
(X , Y , z)
·
-Op
=
(xi y' z') , ,
-
(x y z)
, ,
:
(x' -
X , y' y z' z)
-
,
-
, Equations of Lines :
Q l
i =
% :
p + t
a
i,
a ,
O
=
(a , b ,
c) + +(V ,, Vz Vs) ,
=
(a ++ V
, b+ +Vz , C+ +Vs)
X =
a+ +V,
y =
b + +V
z =
C + tV
ei a (5 j)
a
=
+ + -
P
in = a + +5 -
th
·
O
=
(l -t)a + +
EX finding the eg of a line a direction :
: .
in
ala an as) , , ,
i lorg , in
i =
(9 ,, 92 93) ,
+ ti
CX finding the eg
. of a line between 2 points
:
:
ala ,
az 93) , b(b bribs) ,
v =
5 -
a =
(b , -
a, bz Az
-
, by As)
-
i =
a + t
i =
(9 ,, Az ,
as) + + (b , -
a ,, by 02 , by az)
-
1 2
.
:
The inner product ,
length , a distance
inner product
:
& 5
.
=
a ,b, +
azbz + Asks (scalar)
Properties :
I . . , 0
a c .
=
0 when a j =
x 5 a
-
2 .
. =
x( 5) - =
xb
.
3 & (5 + 5) .
= .
5 + a.
H & 5 .
=
5.
.5 =
11 11 III COSIO)
Proof :
115-all =
11 + 115112 Ill @IllI511 cos(t) -
15-2) (5 - a) .
=
lat + 11b(R-IIIIIIIIICOS(E)
5 5
. -
25 +. =
11 + 115112 -
ZII/IIE11cOSIO)
