Class Notes (939,481)
US (363,169)
Chicago (780)
PHYS (4)
All (3)
Reference Guide

Vectors & Matrices - Reference Guides

4 Pages
976 Views

Department
Physics
Course Code
PHYS 24300
Professor
All

This preview shows page 1. Sign up to view the full 4 pages of the document.
www.permacharts.com
Vectors & Matrices
Vectors & Matrices
3-DIMENSIONAL V E C T O R S
DETERMINANTS DEFINITIONS
VECTOR OPERATIONS
DEFINITIONS
FURTHER GEOMETRIC RELATIONS
© 1996-2013 Mindsource Technologies Inc.
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
||
=×
×
()
Vector Addition
a+b=c implies
111
222 333
+,
 + , +
ab=c
ab=c ab=c
AAddition is
and
commutative
 associative
a
a+b=b+a
(u+v)+w=u+(v+w)
Subtraction
=additiion of
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
:
TM
permacharts

Loved by over 2.2 million students

Over 90% improved by at least one letter grade.

Leah — University of Toronto

OneClass has been such a huge help in my studies at UofT especially since I am a transfer student. OneClass is the study buddy I never had before and definitely gives me the extra push to get from a B to an A!

Leah — University of Toronto
Saarim — University of Michigan

Balancing social life With academics can be difficult, that is why I'm so glad that OneClass is out there where I can find the top notes for all of my classes. Now I can be the all-star student I want to be.

Saarim — University of Michigan
Jenna — University of Wisconsin

As a college student living on a college budget, I love how easy it is to earn gift cards just by submitting my notes.

Jenna — University of Wisconsin
Anne — University of California

OneClass has allowed me to catch up with my most difficult course! #lifesaver

Anne — University of California
Description
Vectors are an important area of study that develops naturally from a basic understanding of physics and is subsets – statics, dynamics, and mechanics. This practical Guide provides assistance in an often complex discipline with powerful graphics and examples that really amplify the topic in a meaningful fashion
More Less
Unlock Document


Only page 1 are available for preview. Some parts have been intentionally blurred.

Unlock Document
You're Reading a Preview

Unlock to view full version

Unlock Document

Log In


OR

Don't have an account?

Join OneClass

Access over 10 million pages of study
documents for 1.3 million courses.

Sign up

Join to view


OR

By registering, I agree to the Terms and Privacy Policies
Already have an account?
Just a few more details

So we can recommend you notes for your school.

Reset Password

Please enter below the email address you registered with and we will send you a link to reset your password.

Add your courses

Get notes from the top students in your class.


Submit