-
(Rectangular Coordinates System in Space) O X Y Z XY X Y YZ Y Z
XZ X Z
4.1
-
8 (octant) X X Y Z
-
P P
-
YZ X X (X-coordinate) P Y Z XZ XY P P (ordered triple) (X,Y,Z)
(cartesian or rectangular coordinates) P (3,2,4) , (4,-2,-5)
(-2,5,3)
-
(3,2,4)
x
y
z
x
y
z
(-2,5,3)
x
y
z
(4,-2,5)
-
2 (right-handed coordinate systems) X Y Z
-
4.1.1
1 1 1 1, ,P x y z 2 2 2 2, ,P x y z
2 2 2
1 2 2 1 2 1 2 1PP x x y y z z
3,4, 1P 2,5, 4Q
2 2 2
2 3 5 4 4 1PQ
25 1 9
35
-
1 1 1 1, ,P x y z
2 2 2 2, ,P x y z1 2 1 2 1 2, y , z
2 2 2
x x y y z zx
-
4.2 (Vector in three dimensional Space) 4.2.1
PQ
P Q P Q
PQ PQ PQ
P
OP
P
-
= 1, 2, 3 1, 2 3 (component)
-
= 1, 2, 3 (directed line segment)
, , + 1, + 2, + 3
-
= 0,0,0
(magnitude) = 1, 2, 3
= 12 + 2
2 + 32
= 1, 2, 3 , , , 0 , (direction angles) , cos cos (direction
cosines)
-
, ,
(4.2.1) =
1
, cos =
2
, cos =
3
-
cos,cos,cos
(direction numbers)
(4.2.1) cos = 1,
cos = 2 cos = 3 1, 2, 3
-
cos, cos, cos 1, 2, 3
cos2 + cos2 + cos2 =1
2
2+
22
2+
32
2
=1
2:22:3
2
2
=1
2:22:3
2
12:2
2:32
2
= 1
cos2 + cos2 + cos2 = 1
-
= 3,2, 6
cos =1
=
3
32:22: ;6 2
=3
9:4:36
=3
7
cos =2
=
2
7
cos =3
=
;6
7
1, 2, 3 = 3,2, 6
-
= 1, 2, 1
= (1)2+( 2)2+ 1 2 = 4 = 2
cos =1
=
;1
2 =
2
3
cos =2
=
2
2 =
4
cos =3
=
1
2 =
3
=2
3, =
4 =
3
-
4.2.2 = 1, 2, 3 ,
= 1, 2, 3 (1) = 1 = 1, 2 = 2 3 = 3 (2) + = 1 + 1, 2 + 2, 3 + 3
(3) = 1, 2, 3
= +
-
= 4,7, 2 , = 8,5,4 3 4
3 4 = 3 4,7, 2 + 4 8,5,4
= 12,21, 6 + 32,20,16= 44,41,10
1 1, 1, 1 2 2, 2, 2 12
(4.2.2)
12 = 2 1, 2 1, 2 1
-
1,3,5 , 2,1,4 1. 2. 3. 1. = (2 1,1 3,4 5) = 1,4,1
2. = (1)2+(4)2+ 1 2 = 18 , , X Y Z cos = 1
18, cos =
;4
18, cos =
1
18
3. 1,4,1
-
, , , 1. + = + 2. + + = + + 3. + 0 = 4. + = 0
5. = =
6. + = + 7. + = + 8. 1 = 9. 0 = 0
-
4.2.3
0
=
1
= 1
-
3,1, 2 2,5,4
= = 2,5,4 3,1, 2 = (1,6,6)
= (1)2+(6)2+ 6 2 = 73
=
(;1,;6,6)
73=
;1
73,
;6
73,
6
73
-
(three basic unit vectors)
= 1,0,0 , = 0,1,0 , = 0,0,1
(basis)
= 1, 2, 3 = 1, 2, 3 = 1, 0,0 + 0, 2, 0 + 0,0, 3 = 1 1,0,0 + 2
0,1,0 + 3 0,0,1 = 1
+ 2 + 3
-
(4.2.2) 12 12 = 2 1, 2 1, 2 1 = 2 1
+ 2 1 +
2 1 (4.2.3)
-
2,1,5 = 2 + + 5 3,4,6 = 3 4 6 8,0, 5 = 8 5
-
4,9,1 2,6,3 1. 2. , , 3. 6,3, 2 = 5
1.
-
2. = 2 4 + 6 9 + 3 1 = 6 3 + 2
3.
=
= 5(;6 ;3 :2
7)
= (;30 ;15 :10
7)
= (;30
7,;15
7,10
7)
-
= (6,3,2) = (2, 2, 2)
2 6 =;30
7; 2 =
;30
7+ 6 =
12
7
2 3 =;15
7; 2 =
;15
7+ 3 =
6
4
2 + 2 =10
7; 2 =
10
7 2 =
;4
7
= (12
7,6
4,;4
7)
-
4.2.4 (dot products) = 1, 2, 3
= 1, 2, 3
= 11 + 22 + 33
-
, 1. = 2. + = +
3. = =
4. = 2
0 = 0 = 0 5. 0 , 0 = cos 0
=
2
= 0
-
= 1,2,3 , = 3,4,2 = 3,6,3 = 0 = 1,2,3 3,4,2 = 3 8 + 6 = 5 =
1,2,3 3,6,3 = 3 12 + 9 = 0 = 3,4,2 3,6,3 = 9 + 24 + 6 = 21
-
4.2.5 (cross products) = 1, 2, 3
= 1, 2, 3
=
1 2 31 2 3
1 2 31 2 3
=2 32 3
1 31 3
+1 21 2
= 23 32 + 31 13
+ 12 21
= 23 32, 31 13, 12 21
-
,
1. =
2. + = +
3. = = 4. = 0 = = 5. = 0 = 0
-
6. = sin 0 7. =
-
= 1,2, 1 = 3,1,2 1. 2.
1. =
1 2 13 1 2
= 4 + 1 2 3 + (1 + 6) = 5 + + 7 1. = = 5 7
-
2. = 25 + 1 + 49 = 5 3
=
1
3 +
1
5 3 +
1
5 3
-
2,1, 1 , 3,0,1 1,3, 2
= = 1,1,2 = = 1,2, 1
-
=
1 1 21 2 1
= 1 22 1
1 2
1 1 +
1 11 2
= 1 4 1 + 2 + 2 1 = 3 +
=1
2
=1
2
=;3 2: ;1 2:12
2
=11
2
-
4.2.6
, , = 1, 2, 3 ,
= 1, 2, 3 = 1, 2, 3
=1 2 31 2 31 2 3
| | ,
-
= = = |cos| = | | = | |
-
= 2 3 , = + = 3 ,
-
4.3 (Line)
1 1 0
1 1
1
-
1 1, 1, 1 , , , , 1
1 =
1
= , , 1, 1, 1 = , , (4.3.1) , , = 1, 1, 1 + , ,
(4.3.1) (4.3.1) (4.3.2) = 1 + , = 1 + , = 1 +
(parametric equation) (4.3.2) , ,
-
= 1
, =
1
, = 1
(4.3.3) 1
=
1
= 1
(4.3.3) (symmetric equations) 1 1, 1, 1 = , , , , , , = 0
(4.3.3)
= 1,;1
,;1
-
, , = = 0 = 1, = 1 = 1 + = 1 = 1
-
1 2,1,1
= 2,3,4
-
3,2, 1 4,4,6 1. 2. 3.
(1) , , = = 4,4,6 3,2, 1 = 1,2,7 3,2, 1 4,4,6 = + , , = 3,2, 1 +
1,2,7
-
= 3 + , = 2 + 2, = 1 + 7
;3
1=
;2
2=
:1
7
(2) 0 1 = 3 + , = 2 + 2, = 1 + 7 = 2 = 5, = 6, = 13 = 1 = 2, =
0, = 8 (3)
;3
1=
;2
2=
:1
7
-
1. = 0 ;3
1=
;2
2=
0:1
7
= 3 +1
7=
22
7 = 2 +
2
7=
16
7
2. = 0 0;3
1=
;2
2=
:1
7
= 4, = 22
3. = 0 ;3
1=
0;2
2=
:1
7
= 2, = 8
-
= 3 + , = 2 + 2, = 1 + 7 = 3 + , = 2 + 2, = 1 + 7 3,2, 1 4,4,6 =
4 + , = 4 + 2, = 6 + 7
;4
1=
;4
2=
;6
7
1,2,7 , 2, 7 1,2,7 = = 3,2, 1 4,4,6 = 1,2,7
-
;1
1=
;2
2=
;4
3
;2
2=
;4
;4=
;7
;6
-
4.4 (Planes) 1
0 1 1
(normal vector)
-
1 1, 1, 1 = + + , , 1
1 = 0 , , 1, 1, 1 = 0 (4.4.1)
1 + 1 + 1 = 0
(4.4.1) 1 1, 1, 1 (4.4.2) = 1 + 1 + 1
+ + + = 0
-
, , (4.4.2) 1 2,1,3 = 2,3,4
-
0 1,2,3 , 1 2,1,1 2 0,1,2 , , 0, 1, 2 01 02 01 02 01 02 01 02 =
01 02 = 01 02 = 1,3,2 1,1,1
=
1 3 21 1 1
= + 3 4
-
0 1,2,3 = 1,3, 4 1 + 3 2 4 3 = 0 + 3 4 = 7
-
2,1,3
=;3
;5=
;1
6=
:2
;2
-
2 3 3 3 3
x y
( , )x y
,x y z
( , , )x y z
4.5
-
G,H,I,J
2 2 2 2
0 0 0( ) ( ) ( )x x y y z z r 2 2 2 2
0 0 0( ) ( ) ( )x x y y z z r
0 0 0( , , )x y z
r
2 2 2 0x y z Gx Hy Iz J
-
3
( * )
X
Y
Z
ax by cz d
, , ,a b c d
( * )
, , ,a b c d
0y z ,0,0d
a
0x z
0x y
0, ,0d
b
0,0,d
c
-
2 2z x y
X
Y
Z
0y z
0x z
0x y
(0,0,2)
( 2,0,0)
(0, 1,0)
z
y
x
-
3
1x y
2 3
(0,1)
(1,0) 0
z
x
y x
y
(1,0,0)
(0,1,0)
-
3
1x y
2 3
X=2
2 0
z
x
y x
y
(2,0,0)
-
3
3
2 2 2
2 2 2
2 2 2
x y a
x z a
y z a
a
-
3
2 2 1x z
2 3
-
3
2y x
2 3
-
(The ellipsoid)
2 2 2
2 2 21
x y z
a b c
2 2 2 0Ax By Cz Dxy Exz Fyz Gx Hy Iz J
-
xy
yz2 2
2 21, 0
y zx
b c
2 2
2 21, 0
x yz
a b
-
xz2 2
2 21, 0
x zy
a c
-
2 2 2
2 2 21
x y z
a b c
xy2 2
2 21, 0
x yz
a b
-
yz
xz2 2
2 21, 0
x zy
a c
2 2
2 21, 0
y zx
b c
-
2 2 24 25 100x y z
2 2 2
125 100 4
x y z
, 0yz x
, 0xy z
, 0xz y
2 2
1100 4
y z
2 2
125 100
x y
2 2
125 4
x z
-
2 2
2 2, 0
x y zc
a b c
, 0z k k
, 0xy z 2 2
2 20
x y
a b
2 2
2 2, 0
x y kk c
a b c
-
, 0yz x
, 0xz y
2
2
y z
b c
2
2
x z
a c
-
, 0yz x
, 0xz y
, 0xy z
2
19
yz
2
125
xz
2 2
125 9
x y
2 2
125 9
x yz
-
2 2 2
2 2 20
x y z
a b c
z k
, 0xy z 2 2
2 20
x y
a b
2 2 2
2 2 2,
x y kz k
a b c
-
2
2
, 0yz x
, 0xz y
2 2
2 20, 0
y zx
b c
2 2
2 20, 0
x zy
a c
-
0x
0z
0y
2 225y z
2 24x y
2 2 24 25 0x y z
5y z
2y x
y k 10y 2 24 25 100x z
