3 eucliden vector
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Three
Euclidean Vector Spaces
3.1 (space)
( , )x y ( , , )x y z 3 (3-space) 3
3 ( Coordinate axes) X XY XZ Y XY YZ Z YZ XZ (Origin) O
3.1.1
1 1 1( , , )P x y z
2 2 2( , , )Q x y z
2 2 2
2 1 2 1 2 1( ) ( ) ( )PQ x x y y z z
n n
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Page 76 | Euclidean Vector Spaces
2 3
3.1.2 , ,x y z ( , , )x y z 3 ( , , )x y z O ( , , )x y z
3.1.3
1 1 1 1( , , )P x y z 2 2 2 2( , , )P x y z
1 2P P
2 1 2 1 2 1
( , , )x x y y z z
3.1.4 3
1 2 3( , , )u u u u
1 2 3( , , )v v v v 3
u v 1 1 2 2 3 3, ,u v u v u v
1 1 2 2 3 3( , , )u v u v u v u v k
1 2 3( , , )ku ku ku ku
(Standard operations) 3
3 0 (0,0,0)
1 2 3( , , )u u u u u
1 2 3( , , )u u u u
3 3.1.5 u , v w 3 ,k m
1. u v 3 2. u v v u 3. ( ) ( )u v w u v w 4. 0 3 0 0u u u 5. u u 3 ( ) ( ) 0u u u u 6. ku 3 7. ( ) ( ) ( )km u k mu m ku 8. ( )k u v ku kv 9. ( )k m u ku mu 10. 1u u
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infinite dimension n n -space 2 3-space
n (Vectors in n-Space) 2-3 3 () n 3.1.6 n n (ordered n-tuple) n
1 2( , ,..., )
na a a n
n (n-space) n
1 2
( , ,..., )n
a a a n n 3.1.7
1 2( , ,..., )
nu u u u
1 2( , ,..., )
nv v v v n
u v 1 1 2 2, ,..., n nu v u v u v
1 1 2 2( , ,..., )n nu v u v u v u v k
1 2 3( , ,..., )ku ku ku ku
(Standard operations) n
3 0 (0,0,...,0)
1 2
( , ,..., )n
u u u u u
1 2( , ,..., )
nu u u u
3 n
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Page 78 | Euclidean Vector Spaces
n Properties of vector Operations in n-Space 3.1.8 u , v w n ,k m
1. u v n 2. u v v u 3. ( ) ( )u v w u v w
4. 0 n 0 0u u u 5. u u n ( ) ( ) 0u u u u
6. ku n 7. ( ) ( ) ( )km u k mu m ku 8. ( )k u v ku kv 9. ( )k m u ku mu 10. 1u u
3.1.9 v n k 1. 0 0v 2. 0kv 3. 1v v
(Linear Combinations) 3.1.10 w
n w 1 2, ,..., nv v v n
1 1 2 2...
n nw k v k v k v
1 2, ,...,
nk k k
1 1 2 3, ,c c c
1 2 3(1, 1,0) (3,2,1) (0,1,4) ( 1,1,19)c c c
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3.1 1. (1,3,3)u , ( 1,0,3)v (1, 2,3)w
a. 2u v
b. u w v
c. 2 3v w 2. 1 x 2 7u v x x w 3. , ,a b c ( 2,9,6) ( 3,2,1) (1,7,5) (0,5,4)a b c 4. , ,a b c (1,2,0) (2,1,1) (0,3,1) (0,0,0)a b c
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Page 80 | Euclidean Vector Spaces
3
1. (Inner Product) 2. ( Outer
Product) 3. (Cross Product)
3 () ( Outer Product) ( Dyad Product)
3.2
3.2.1
1 2( , ,..., )
nu u u u
1 2( , ,..., )
nv v v v n
(Euclidean inner product) u v u v
1 1 2 2
1
...n
i i n ni
u v u v u v u v u v
2n 3n (dot product)
- 2u u u - Tu v v u
(Properties of Euclidean inner product) 3.2.2
1 2 1 2( , ,..., ), ( , ,..., )
n nu u u u v v v v
1 2( , ,..., )
nw w w w
n ,k m 1. u v v u 2. ( )u v w u v u w 3. ( ) ( ) ( )k u v ku v u kv 4. ( ) ( ) ( )ku mv km u v 5. 0u u 0u u 0u
1 2
( , ,..., )n
u u u u , 1 2
( , ,..., )n
v v v v
1 2
( , ,..., )n
w w w w n ,k m 1. u v v u 2. ( )u v w u v u w
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3
3. ( ) ( ) ( )k u v ku v u kv 4. ( ) ( ) ( )ku mv km u v 5. 0u u 0u u 0u
3.2.3
1 2( , ,..., )
nu u u u n u
u
2 2 2 21 2
...n
u u u u u u u
3.2.4
1 2( , ,..., )
nu u u u
1 2( , ,..., )
nv v v v n
( Euclidean distance) u v d( , )u v
d( , )u v u v
2 (2,3, 1,7), (1,0,3, 5)u v d( , )u v
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Page 82 | Euclidean Vector Spaces n () (norm) ( Euclidean norm)
Cauchy-Schwarz 3.2.5 Cauchy-Schwarz
1 2( , ,..., )
nu u u u
1 2( , ,..., )
nv v v v n u v u v
3.2.6 u v
n k
1. 0u
2. 0u
0u
3. ku k u
1 2
( , ,..., )n
u u u u 1 2
( , ,..., )n
v v v v n k 1. 0u
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2. 0u 0u
3. ku k u 3.2.7 u v
n k u v u v
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Page 84 | Euclidean Vector Spaces
3.2.8 Parallelogram Equation for Vectors
u v n
2 2 2 22u v u v u v
3.2.9 u v
n 2 21 1
4 4u v u v u v
3.2.10 ,u v w
n k 1. d( , ) 0u v
2. d( , ) 0u v u v 3. d( , ) d( , )u v v u 4. d( , ) d( , ) d( , )u w u w w v
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3.2 1.
a. ( 4,3)u b. (2,2,2 2)v c. (5,6,0)w
2. (unit vector)
a. u n u
b. u n u 3. (1, 1, 4,2,3)u (2,3,4,5,6)v 4. ,u v w
n k (1.) d( , ) 0u v (2.) d( , ) 0u v u v (3.) d( , ) d( , )u v v u (4.) d( , ) d( , ) d( , )u w u w w v
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Page 86 | Euclidean Vector Spaces
3.3 (Orthogonality) (dot product) 2-3
cosu v u v
u v n
cosu v
u v
3.3.1 u v
n 0u v
3 ( 2,3, 1,4), (1,2,0, 1)u v 3.3.2 u v
n 2 2 2
u v u v
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3.4 3 L 3
0 0 0 0( , , )P x y z
( , , )a b cv L ( , , )P x y z L 0P P
v t
0PP t v
0 0 0( , , ) ( , , )x x y y z z ta tb tc
0 0,x x ta y y tb 0z z tc
3.4.1 3 , ,a b c
0 0 0, , ( )x x ta y y tb z z tc t
0 0 0 0( , , )P x y z ( , , )a b cv
0 0 0
, , ( )x x ta y y tb z z tc t 1
0(2,2, 3)P
(4,5, 7) v 2 (2,4, 1)A
(5,0,7)B XY
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Page 88 | Euclidean Vector Spaces
( , , )x y zr ( , , )P x y z 0 0 0 0( , , )x y zr
0 0 0 0( , , )P x y z 0PP 0r r
t 0
r r v t
0r r v
0 0 0x x y y z z
a b c
3.4.2 n
0r v 0
n t t
0r r v
0r v
3
0( 1,2)P ( 2,3) v
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4
0(1,2, 3)P (4, 5,1) v
5
4 11 393 7 26
2 8 26
x y z
x y z
x z
6 3
4
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Page 90 | Euclidean Vector Spaces
( inclination) (normal vector)
0 0 0 0( , , )P x y z ( , , )a b cn
(normal vector) ( , , )P x y z
0P P
n
00PP n
0 0 0( ) ( ) ( ) 0a x x b y y c z z
point-normal form
3.4.3 3 , ,a b c
0ax by cz d ( , , )a b cn (normal vector)
0ax by cz d , ,x y z 7 (1, 2,5)
(4,2, 3) n
3 (normal vector) 3.4.4 ( cross product vector product)
1 2 3( , , )u u u u
1 2 3( , , )v v v v u v
2 3 3 2 3 1 1 3 1 2 2 1( , , )u v u v u v u v u v u v u v
3.4.5 u v 3 u v u v
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8 1 2(1,2, 1), (2,3,1)P P
3(3, 1,2)P
( , , )x y zr ( , , )P x y z 0 0 0 0( , , )x y zr
0 0 0 0( , , )P x y z 0PP 0r r
1 2, v 0 v 0
1 1 2 2t t
0r r v v
3.4.6 n
0r 1 2, v 0 v 0
n 1 2,v v
1 2,t t
1 1 2 2t t
0r r v v
0r 1 2,v v
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Page 92 | Euclidean Vector Spaces
9 9 5 16 0x y x
10 2 5x y z
11
9 5 1618 2 10 32
27 3 15 48
x y z
x y z
x y z
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12 1z y z
2 3 1z y z
13 1z y z
2 3 1z y z
3.4.7
0 0 0 0( , , )P x y z 0ax by cz d D
0 0 0
2 2 2
ax by cz dD
a b c
14 (1, 4,3) 2 3 6 1 0x y z
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Page 94 | Euclidean Vector Spaces
3.4 1. P u
a. ( 4,1), (0, 8)P u
b. ( 9,3,4), ( 1,6,0)P u
2. (4,0, 5)u
3.
3 2 06 4 2 0
3 2 0
x y z
x y z
x y z
3 2 26 4 2 4
3 2 2
x y z
x y z
x y z
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2( )f x x ( real valued function of real variable) x
2 2( , )f x y x y ( real valued function of two real variables)
4.4 n m
n
2( )f x x 2 2( , )f x y x y 2
2 2 2( , , )f x y z x y z 3 n m f n m f
(transformation map) n m : n mf 1m m n
: n nf n (operator n )
1 2 3, , ,..., mf f f f n
1 1 1 2
2 2 1 2
1 2
( , ,..., )
( , ,..., )
( , ,..., )
n
n
m m n
w f x x x
w f x x x
w f x x x
1 2( , ,..., )nx x x n 1 2( , ,..., )mw w w
m m T
: n mT
1 2 1 2( , ,..., ) ( , ,..., )n mT x x x w w w
15
1 1 2
2 1 2
3 1 2
w x x
w x x
w x x
2 3 2 3:T 1 2 1 2 1 2 1 2( , ) ( , , )T x x x x x x x x (1,2) (3,2, 1)T
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Page 96 | Euclidean Vector Spaces Some Notational Matters
A T
( )AT Ax x
T
( )T T x x
AT A
n m 1 2 1 2( , ,..., ) ( , ,..., )n mT x x x w w w 1 2 3, , ,..., mf f f f
1 1 1 2
2 2 1 2
1 2
( , ,..., )
( , ,..., )
( , ,..., )
n
n
m m n
w f x x x
w f x x x
w f x x x
: n mT n m
: n mT
1 11 1 12 2 1
2 21 1 22 2 2
1 1 2 2
...
...
...
n n
n n
m m m mn n
w a x a x a x
w a x a x a x
w a x a x a x
1 11 12 1 1
2 21 22 2 2
1 2
n
n
m m mn nm
w a a a xw a a a x
a a a xw
Aw x
ijA a T
16 4 3:T
1 1 2 3 4
2 1 2 3 4
3 1 2 3 4
2
3
4
w x x x x
w x x x x
w x x x x
11
22
33
4
2 1 1 13 1 1 14 1 1 1
xw xw xw x
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1 2 (operator) y
x
y x
2 3 (operator) xy
xz
yz
2 3
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Page 98 | Euclidean Vector Spaces
3 (projection) 2 (operator) (orthogonal projection) x
(orthogonal projection) y
4 (projection) 3 (operator) (orthogonal projection) xy
(orthogonal projection) xz
(orthogonal projection) yz
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5 2 (operator)
6 3 (operator) x
y
z
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Page 100 | Euclidean Vector Spaces
7 2 (operator)
k 2 (0 1)k
k 2 ( 1)k
7 3 (operator)
k 3 (0 1)k
k 3 ( 1)k
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4.3 1. A T
3 2:T
(1,0,0) (7,11), (0,1,0) (6,9)T T (0,0,1) ( 13,17)T
2. 2 3:T
11 2
2
1 4
2 5
3 6
xT x xx
3. 1 2, , , n
mv v v : m nT
1
2
1 1 2 2 m m
m
x
xT x v x v x v
x
(
1 2, , ,
mv v v )
4. 2 2:T 45 T
5. 2 2:T T
6.
1 1 22 1 2
7
3 20
y x x
y x x
7. A T (linear operator)
a. 3 0
0 3A
b. 1 0
0 1A
c. 1 1
1 1A
d. 1 0
0 0A
e. 0 2
2 0A
f. 0.8 0.6
0.6 0.8A
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Page 76 | Euclidean Vector Spaces
8. ( )T x Ax
a. 0 1
1 0A
b. 2 0
0 2A
c. 1 0
0 2A
d. 0 0
0 1A
e. 0.6 0.8
0.8 0.6A
1
1
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T:R4 R3 T(a1,a2,a3,a4) =(a1,a1+a3,a3+a4) S4 , S3 R
4 ,R3 S = {(1,0,0,1),(0,0,0,1),(1,1,0,0),(0,1,1,0)} , S 1 ={(1,1,0),(0,1,0),(1,0,1)} R4 , R3 T S S3 T S S 1 T(2,1, 1,3) 1.1 1.2 T T:R2 R2 T(x,y) =(x+2y,2x3y) T {(1,0),(0,1)} {(2,1),(1,0)} {(1,1),( 2,3)} {(1,1),(0,2)} T:R2 R3 T(x,y) = (x+y , 2xy , x+2y) T 3.1 {(1,1),( 1,2)} {(1,0,0),(1.0.1),(0,1,1)} 3.2 {(1,0),(0,1)} {(1,0,1),(0,1,1)(1,1,0)} 4. A B 4.1 A = {(1,0),(0,1)} ; B = {(1,1),( 1,2)} 4.2 A = {(1, 1),(2,1)} ; B = {(1,2),(3,3)} 4.3 A = {(1,1),( 1,2)} ; B = {(1,0),(0,1)} 4.4 A = {(1,2),( 3,1)} ; B = {(1, 1),(2,1)}
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