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Reference Guide

# Vectors & Matrices - Reference Guides

4 pages525 viewsFall 2015

Department
Physics
Course Code
PHYS 24300
Professor
All
Chapter
Permachart

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Vectors & Matrices
Vectors & Matrices
3-DIMENSIONAL V E C T O R S
DETERMINANTS DEFINITIONS
VECTOR OPERATIONS
DEFINITIONS
FURTHER GEOMETRIC RELATIONS
VECTORS & MATRICES • A-863-X1
EIGENVA L U E S & E I G E N V E C T O R S
a
aijkau
a
=(=vector)
=++ =||
=
where 
12 3
PQPQ
aa a
 | a | [ i j kcos + cos + cos
[,,]=
12 3
123
αα α]
aa a vvector components
Z
a
a
Q
Y
X
P
1a2
a3
General
||
=++
1
2
2
2
3
2
aa aaa
Magnitude or Length, | a |
Unit Vector, u
uijkaa
i, j, k
=cos +cos +cos = | |
=
12 3
unit
α α α
 vectors along  ,  ,  -  axes--x y z
Directional Cosines
[cos , cos , cos ]
=[cos( ),cos( ),
123
ααα
i,a j,a  cos( )]
cos + cos + cos = 1
2
1
2
2
2
3
k,a
αα α
Z
a
Y
X
a3
α
3
α
2
α
1k
a2j
a1i
Position Vector, r
rijkru=++ =
||
xyz
Z
k
Y
X
ij
u
r
A
y
x
z
Z
YX
Parallel Vectors
Two non -zero vectors are parallel if
=a b×00orsin( )=0a,b
Coplanar Vectors
dAP AB PQ
AB PQ
1
||
=×
×
()
dAP BP CP
AB AC
2
||
=×
×
()
a+b=c implies
111
222 333
+,
 + , +
ab=c
ab=c ab=c
and
commutative
 associative
a
a+b=b+a
(u+v)+w=u+(v+w)
Subtraction
negativevector
a
Y
X
a2
b2
c2
b
c
b1
a1c1
Vector (Outer or
Cross Product)
aabbcc
aa,,bb
×==
×=
vectornormalto
|a b| |a||bb|
[(a a)(b b) – (a b) ]
sin
area of para
α
=
=
⋅⋅ ⋅
21/ 2
lllelogram
2 times area of triangle
=
ab
i
×=
jjk
aa a
bb b
123
12 3
Cross product is distri bbutive but
associative or commutati
not vve
  aaaa aacc
   aaaa cc
   aa
×+ ×+×
×××+
(b c) b
(b c) b
=
××=×bbaa
=− +
−+
(a b a b
(a b a b
(
23 3 2
3 1 1  3
)
)
i
j
aab ab
1 2 2  1
)k
Multiplication by Scalar
Quantity, q
Length:
Direction: same as
|a|= | | |a|
a
qq
if 0;
oppositeif 0
Multiplicatio
q
q
>
<
nnbyascalariscommutative,
associative and distributive
= qqpq pqqaa(a)()a= q q(a)+=+bbaa bb
a2aaa
1
2
Scalar (Inner or Dot) Product
ab |a||b|==cos a scalar (i.e., a numberα))
11 2 2 3 3
=ab+ab +ab
=|a| b a(Projection, , of on )p
=
=
|b| a b
ab a
(Projection, ,of  on )
0 if
p
,,orthogonal
(also:if =0or =0)
b
ab
aa
===
=
⋅⋅
⋅⋅
|a|
ij=kj=ik
ii=jj
2
1
2
2
2
3
2
0
a+a+a
==k k
 
=1
Dot product is commutative
and d
ii stributive
++
a b=b a
(a b) c=a c b c
⋅⋅
⋅⋅⋅
b
a
α
Projection, p > 0
a
α
b
b
a
α
Projection, p = 0
Projection, p < 0
Repeated Product:
Scalar Triple (Box)
Product (= a scalar)
b
a
α
h
b c
x
c
a(b c) |a||b c|×= = ×
aa a
bb b
cc c
123
12 3
123
cos
α
= signed  pparallelepiped volume
with sides
Equ
a,b,c
ii valent scalar triple products
a(b c)=(a
××bb)c=b (a c)=
b(c a)=c(a b)
⋅⋅
⋅⋅ ×
××
Repeated Product: Vector Triple Product
(=a vector)
abc
ij k
××=aa a
bc
12 3
21
00-
=(a c)b–(a b)c=
abci abcj
221 121
⋅⋅
–
DEFINITIONS EVALUATION
PROPERTIES
Determinant
 
  D=
213
486
075
hass=3
n
det A= |A| = D = a numerical (or functional) measure of
the square matrix A
Order n of a Determinant
Minor Mij of Entry aij in Matrix A
Determinant of
submatrix
w
and
c 
Cofactor Aij of Entry aij in Matrix A
AAMMAA3322iijjiijj
=(1) D=
213
486
075
has = – 23
46
i+j
• The set of values liwhich satisfy the
vector equation Ax = lxare termed
eigenvalues of the (square) matrix A
• The corresponding column vectors xi
are known as the eigenvectors of the
(square) matrix A
• The eigenvalues of Aare the
roots of the characteristic
equation det [AlI] = 0
• The eigenvectors xiof Aare
obtained by substitution of
the eigenvalues into the
vector equation: (AliI) = 0
Every n xnm atrix has n(real or complex) eigenvalues • All eigenvalues
of a symmetric matrix are real
In a triangular or diagonal matrix, the eigenvalues, l1liare identical
to the diagonal entries a11, ann
The sum of the diagonal entries of any square matrix Aequals the sum
of the eigenvalues of the matrix • This sum is known as the trace of
A= tr A= Âli
The product of the eigenvalues of a matrix equals the determinant of the
matrix: li= det A
Distance d1Between Two
Lines AB and PQ
Distance d2of a Point, P,
from a Plane (A, B, C)
b
a
a b
x
b a =
xa b
x
α
a
Y
X
a2
b2
c2
b
c
b1
a1c1
a
ab
a
ba
Number of rows
or columns in D D=
213
486
075
hass=3
n
Determinant of
submatrix
without
row  and
col
i
uumn
D= 213
486
075
has = 23
46
j
MM3322
Determinant of
submatrix
without
row  and
col
i
uumn
D= 213
486
075
has = 23
46
j
MM3322
Determinant of
submatrix
without
row  and
col
i
uumn
D= 213
486
075
has = 23
46
j
MM3322
:
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