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MATEMATIKA TEKNIK KIMIA I

Vektor dan Aplikasinya

Siswo Sumardiono

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Unit Vectors in Rectangular

Coordinate Systemy

x

z

ik

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Vector Representation:

x y z A A A= + +A i j k

The unit vectors , , and should not bei j k

2 2 2

Magnitude or Absolute Value:

x y z A A A A= = + +A

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The three basic laws of algebra obeyed by any given vector

A, B, and C, are summarized as follows:

Law Addition Multiplication

Commutative

Associative

ABBA+=+

C)BA()CB(A ++=++

kAAk=

A)kl()Al(k =

Distributive

where k and l are scalars

BkAk)BA(k +=+

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Example. A force has x, y, and zcomponents of 3, 4, and 12 N,

respectively. Express the force as a vector inrectangular coordinates.

3 4 12= + F i j k

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Example Determine the magnitudeof the force in Example before.

3 4 12= + F i j k

2 2 2(3) (4) ( 12)

13 N

F= + +

=

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Vector Operations to be Considered

Scalar or Dot Product AB

Vector or Cross Product AxB

Triple Scalar Product (AxB)C

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When two vectors and are multiplied, the result is

either a scalar or a vector depending on how they aremultiplied. There are two types of vector multiplication:

1. Scalar (or dot) product:

2.Vector (or cross) product:

A

BA

B

A

BA

B

The dot product of the two vectors and is definedgeometrically as the product of the magnitude of and the

projection of onto (or vice versa):

where is the smaller angle between and

ABcosABBA =

AB

A BB

A B

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Consider two vectors A and B

oriented in different directions.

B

A

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Scalar or Dot Product

Definition:

cosAB =A B

Computation:

x x y y z z

A B A B A B= + +A B

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First Interpretation of Dot Product:

Projection ofA on B times the length ofB.

A

cosA

B

B

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Second Interpretation of Dot Product:

Projection ofB on A times the length ofA.

A

cosB

B

B

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Some Implications of DotProduct0

The vectors are parallel to each other and

=

=

o

90

The vectors are to each other and0

=

=A B

o

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Example. Perform several scalar

operations on the following vectors:2 2= +A i j k

3 4 12= + +B i j k2 2 2

A A A= + +

2 2 2(2) ( 2) (1) 3

x y z

= + + =

2 2 2

2 2 2

(3) (4) (12) 13

x y z B B B B= + +

= + + =

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Example. Continuation.

(2)(3) (-2)(4) (1)(12) 10

x x y y z zA B A B A B= + +

= + + =

A B

10 10cos 0.2564

3 13 39AB

= = = =

A B

1 ocos 0.2564 75.14 1.311 rad = = =

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If and then

which is obtained by multiplying and component bycomponent

),A,A,A(A ZYX= )B,B,B(B ZYX=

ZZYYXXBABABABA ++=

A B

ABBA =

Review of VectorReview of VectorAnalysisAnalysis

CABACBA +=+ )(

A A = A2

= A2

eX ex = ey ey = eZ ez = 1

eXey

=ey

ez

=eZ

ex

=0

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Vector or Cross Product

( )

Definition:

sinAB = nA B u

omputat on:

x y z

x y z

A A

B B B

=

i j k

A B

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Cross Product AxB

B

A

A B

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Cross Product BxA

B

A

B A

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Area of parallelogram below is the

magnitude of the cross product.

B

A

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Some Implications of Cross

Product0

The vectors are parallel to each other and

0

=

=A B

o

( )

90

The vectors are to each other andAB

=

= nA B u

o

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Note that the cross product has the following basic

properties:(i) It is not commutative:

It is anticommutative:

ABBA

ABBA =

(ii) It is not associative:

(iii) It is distributive:

(iv)

CABACBA +=+ )(

0AA = )0(sin =

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If and then

zyx

zyx

AAA

eee

BA =

),A,A,A(A ZYX= )B,B,B(B ZYX=

zyx

zxyyxyzxxzxyzzy e)BABA(e)BABA(e)BABA( ++=

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2 2 1 x y z

A A A= =

i j k i j k

A B

Example. Determine the cross

product of the vectors

x y z

[ ] [ ]

[ ]

( 2)(12) (1)(4) (2)(12) (1)(3)

(2)(4) ( 2)(3)

28 21 14

=

+

= +

A B i j

k

i j k

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Example Determine a unit vectorperpendicular to the vectors

2 2 2( 28) ( 21) (14) 37.70= + + =A B

28 21 1437.70

0.7428 0.5571 0.3714

+= =

= +

nA B i j kuA B

i j k

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Triple Scalar Product

Definition:

( ) A B C

( )

x y z

x y z

x y z

A A A

B B BC C C

=A B C

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Volume of parallelepiped below is

the triple scalar product of thevectors.

B

A

C

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Example Determine the triple

scalar product of the vectors2 2= +A i j k

3 4 12= + +

B i j k3 5 6= + C i j k

( ) 3 4 12

3 5 6

2( 24 60) 2( 18 36) (15 12)

168 108 3 273

x y z

x y z

x y z

B B B

C C C

= =

= + +

= + =

A B C