MTKI 221211 Vektor
Transcript of MTKI 221211 Vektor
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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