review of mathematics. review of vectors analysis givenmagnitude of vector: example: dot product: ...
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REVIEWOF
MATHEMATICS
Review of Vectors Analysis
GivenT
zyxT
zyx bbbbaaaa ],,[,],,[
Magnitude of vector:222zyx aaaaaa
Example: 14321]3,2,1[ 222 aa T
Dot product: )cos(babababababa zzyyxxT
is the angle between the two vectors.
Example:
563544
cos56
35
44)6(5)4(3)2(1]6,4,2[
]5,3,1[
1b
a
bab
aT
T
Review of Vectors Analysis
Vectors and are said to be perpendicular or orthogonal if
a
b
0ba
Example:
baba
ba TT
0
010,001
Note that the above vectors represent the unit vectors for the X-axis and Y-axis. They are definitely perpendicular or orthogonal.
Cross product:
)sin(
ˆˆˆ
det321
baba
bbb
aaa
eee
ba
bababababababa
zyx
zyx
xyyxzxxzyzzy
is the angle between the two vectors.
Review of Vectors Analysis
563524
sin
24
56
35
2)4()2(
ˆ)2(3)4(1ˆ)6(1)2(5ˆ)4(5)6(3
]6,4,2[
]5,3,1[
1
321
ba
b
a
eeeba
b
aTT
T
Example:
Review of Vectors Analysis
The cross product of and provides us with a vector which is perpendicular to both and
a
b
b
a
Example:
1
0
0ˆ)1(ˆ)0(ˆ0
010
001
ˆˆˆ
det
010,001
kji
kji
ba
ba TT
Note that the above vectors represent the unit vectors for the X-axis and Y-axis respectively. Their cross product is the unit vector for the Z-axis, which is definitely perpendicular to both the X-axis and the Y-axis.
Review of Vectors Analysis
Note that the unit vectors for the right handed Cartesian reference frame are orthonormal basis vectors, i.e.
1ˆˆˆ
0ˆˆˆˆˆˆ
ˆˆˆˆˆˆ,ˆˆˆ
kji
ikkjji
jikikjkji
,2,1,0,20)sin(
,2,1,22
0)cos(
nnbaba
nnbaba
Vector triple product: baccabcba
Review of Vectors Analysis
Example:
000
0
1
0
0
1
0
100
001
010
cba
c
b
a
T
T
T
000)000(100
000)000(001
T
T
bac
cab
Review of Vectors Analysis
Scalar triple product:
zyx
zyx
zyx
ccc
bbb
aaa
cbacbacba
det
Example:
10)01(10
100
001
010
det
100
001
010
cba
c
b
a
T
T
T
1100100
1010010
TT
TT
cba
cba
Review of Vectors Analysis
Tzyx aaaadtd
GivenT
zyx aaaa ],,[
Example:
T
T
tadtd
ttta
212
122
Tzyx aaaadtd
)(
where is a any constant
GivenT
zyx aaaa ],,[
Example: T
T
tadtd
ttta
21233
122
Review of Vectors Analysis
bababadtd
GivenT
zyxT
zyx bbbbaaaa ],,[,],,[
Example:
2106}122{)}2(1)64{(
1
0
2
12
1
32
2
1
2
132
12
222
2
2
tttttttbadtd
t
t
t
t
tt
badtd
ttb
tttaT
T
bababadtd
Review of Vectors Analysis
GivenT
zyxT
zyx bbbbaaaa ],,[,],,[
Example:
)sin()cos(11)sin(5
)}sin(3)cos({
)cos(3)sin(
)cos()sin(5
)cos(
)sin(
)}sin()cos(10{
)sin(3
)cos(3
0)sin()cos(
5
ˆˆˆ
det
0)cos()sin(
3110
ˆˆˆ
det
0
)sin(
)cos(5
0
)cos(
)sin(
3
1
10
0)cos()sin(
5
2
23
23
2
3
3
2
2
322
3
2
2
32
ttttt
tttt
tttt
badt
d
tttt
tt
tt
ttt
tt
tt
badt
d
tt
ttt
kji
tt
tt
kji
badt
d
t
t
t
t
t
t
t
t
t
badt
d
ttb
tttaT
T
bAbAbAdtd
Review of Vectors Analysis
where A is a matrix of dimension comparable to the vector being multiplied
Given Tzyx bbbb ],,[
Example:
34
2
16
4
14
3
2
0
1
2
302
30
012
1300
010
002
300
010
002
302
30
012
0121
2222
2
t
t
t
t
t
t
t
tt
t
t
t
t
t
bAdt
d
Adt
d
t
t
t
A
tbdt
dttb TT
Eigenvalues and Eigenvectors
Let A be an nn matrix. If there exists a and a nonzero n1 vector such that
xxA
x
then is called an eigenvalue of A and is called an eigenvector of A corresponding to the eigenvalue
x
Let In be a nn identity matrix. The eigenvalues of nn matrix A can be obtained from:
0)det( nIA
A nn matrix A has at least one and at most “n” distinct eigenvalues
Example 1: Eigenvalues and Eigenvectors
Find the eigenvalues of
700
240
321
A
7,4,1
0741
0
700
240
321
det0)det(
700
240
321
100
010
001
700
240
321
n
n
IA
IA
Solution:
Example 2: Eigenvalues and Eigenvectors
What is the eigenvector of at =1?
36329
4948
33364
491
A
0
0
0
13329
44048
333645
49
1
0
0
0
)4936(329
4)499(48
3336)494(
49
1
0
0
0
100
010
001
1
36329
4948
33364
49
1
0
3
2
1
3
2
1
3
2
1
x
x
x
x
x
x
x
x
x
xIAxxA
013329
044048
0333645
321
321
321
xxx
xxx
xxx
2
09819606516045
0333645
2
3
32321
321
x
x
xxxxx
xxx
Multiply 3rd eqn by -5 and add it to 1st eqn to eliminate 1x
Example 2: Eigenvalues and Eigenvectors
Divide 2nd eqn by and simplify using the known result:2x
32
4832
0)2(44048
044048
2
1
2
1
2
3
2
1
xx
xx
x
x
xx
Example 2: Eigenvalues and Eigenvectors
013329
044048
0333645
321
321
321
xxx
xxx
xxx
T
T xxxxx
xx
x
x
2132
32
,2
2321
2
1
2
3
TT
n
n
x
x
x
xx
x
63271
2132
73
2132
2132
222
2
2
Example 2: Eigenvalues and Eigenvectors
Story so far:
We can obtain a normalized eigenvector using:
Trigonometric Functions
1)(cos)(sin 22
)tan()tan(
)cos()cos(
)sin()sin(
)tan()tan(1)tan()tan(
)tan(
)sin()sin()cos()cos()cos(
)sin()cos()cos()sin()sin(
21
2121
212121
212121
Trigonometric Functions
)sin()sin()cos(
)cos()cos()sin(
dtd
dtd
dtd
dtd
)sin()cos(
)cos()sin(
dddd
)sin()cos(
)cos()sin(
d
d
)cos()sin(
)tan(