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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

© 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×00orsin( )=0a,b

• If represent thena,b,c AB, AC , AD A, B,C,D

lieinaplaneprovided: ( )= abc

⋅⋅ ×00

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

negativevector

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

123

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

(

23 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

>

<

• nnbyascalariscommutative,

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

ha⇒ss=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 [A– lI] = 0

• The eigenvectors xiof Aare

obtained by substitution of

the eigenvalues into the

vector equation: (A– liI) = 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, l1… liare 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 =

x–a b

x

α

→

→

→

→

→

→

→

→

→

→

→

→

→

→

→

→

→

a

Y

X

a2

b2

c2

b

c

b1

a1c1

a

–ab

–a

b–a

Number of rows

or columns in D D=

213

486

075

ha⇒ss=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

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