bab 3 ba501 vector dan scalar
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ENGINEERING MATHEMATICS 4 VECTORBA501
CHAP 3 - Vector and Scalar Products
Vector Quantities
a quantity that has magnitude and direction
Examples of vector quantities are displacement, velocity, direction, momentum,
force, lift, weight and etc.
Scalar Quantities
a quantity that has magnitude only.
Typical examples of scalar quantities are time, speed, temperature, and volume. A
scalar quantity or parameter has no directional component, only magnitude. For
example, the units for time (minutes, days, hours, etc.) represent an amount of time
only and tell nothing of direction. Additional examples of scalar quantities are
density, mass, and energy.
Fundamental of vector
i) Vector notation
vector notation is how to present vector, such us :a) vector is usually given a bold letter, such as A
b) place a right-handed arrow over the letter to denote a vector
c) vector can be write in engineering notation and matrix notation
ii) Vector representation
Vectors can be graphically represented by directed line segments
example : vector AB = a
a
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A
B
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ENGINEERING MATHEMATICS 4 VECTORBA501
If two vectors are represented by two adjacent sides of a parallelogram, then the
diagonal of parallelogram through the common point represents the sum of the
two vectors in both magnitude and direction.
Q Q R
O P O P
OP + OQ = OR
Q Q R
O P O P
OP + OQ = QP Figure 2
iii) Method of component
Rules of vector components:
i) Components should be perpendicular is called the orthogonal
components.ii) The component s of the vector may be in any axis (x and y axis) we called
the horizontal or the vertical dimension.
iii) The direction of the components is look like head to tail, so that we canadd that vector.
iv) If we are adding those x and y vectors we can get the resultant vector.
The components of a vector are those vectors which, when added together,give the original vector.
N
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ENGINEERING MATHEMATICS 4 VECTORBA501
E
The sum of the components of two vectors is equal to the sum of these two
vectors.
A2 A
A = A1 + A2
A1
A2 A
A1
A1, the component in an easterly direction, will have a magnitude
A1 = A cos .
A2, the component in a northerly direction, will have a magnitude
A2 = A sin
Substraction
Subtraction is considered an addition process with one modification that the secondvector (to be subtracted) is first reversed in direction and is then added to the firstvector.
B B
=
O A O A
OA + BA− = OB
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A B C
a
ENGINEERING MATHEMATICS 4 VECTORBA501
Characteristic of additional vectors.
Characteristic Resultant
i. Commutative law a + b = b + a
ii. Associative law (a + b )+ c = a + (b + c )
iii. Identity law a + 0 = a
iv. Inversion law a + (-a ) = 0
Multiplication vector with scalar Multiplication vector , a with scalar value , t produce vector t a where
magnitude |t|.
Characteristic Resultant
i. Commutative law ma = amexp: 2a = a 2
ii. Associative law m(na ) = (mn)a exp : 2x(3a ) = (2x3) a = 6a
iii. Distributive law (m+n)a = ma + na
exp : (2 + 3) a = 2a + 3a = 5a
iv. Distributive law m(a + b ) = ma + mb
exp : 2 ( a + b ) = 2a + 2b
Definition 1 :
Given 3 point A, B, and C. Point A, B and C is collinear if AC t AB = , t is scalar non zero
t a
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ENGINEERING MATHEMATICS 4 VECTORBA501
example 1 : Find the scalar value λ , for equation below :λ OC = OA6 + BC 12 + AO4 + AB2 + OB10
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ENGINEERING MATHEMATICS 4 VECTORBA501
a) MD b ) DL c) LM
4. Given position vector point A, B and C respectively are 4 a + 2b , 8a - 4b
, and 16a - 16b . Show that point A, B and C are collinear and find
ratio of AB : BC .
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ENGINEERING MATHEMATICS 4 VECTORBA501
If a~ = b~
hence a~ • a~ = aa
~~ =2~
a = a2 ; ∴ a~ • a~ = a2
Refer to fig 2,a~and b~ are parallel but in the opposite direction,
hence θ = 180°;
∴ , ba~~ = kosba
~~ 180° = ba~~ (-1) = - ba
~~
3. Perpendicular vector
If a~ and b~ are perpendicular, hence θ = 90°.
∴ a~ • b~ = ba~~ cos 90° = ba
~~ 0 = 0
∴ a~ • b~ = 0
4. Angle between two vector.
Theorem;
; a~ • b~ = kosba~
~θ ⇒ kos θ =
ba
ba
~~
~•~
51
b
~
a~
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ENGINEERING MATHEMATICS 4 VECTORBA501
4 :Given vector a~and b~ respectively a~ = 4 , b
~
=3 and a~ • b~= 7. Find
the magnitude of (a~+ b~ ) and the angle between a
~ and b~ .
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ENGINEERING MATHEMATICS 4 VECTORBA501
5 : Calculate the work done F · s given |F|, |s| and θ (the angle between the
force F and the displacement s) when
|F| = 4 N, |s| = 2 m, θ= 27o
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Exercise : Calculate a~ • b~ if given | a~ | = 3 , | b~ | = 5 and angle between a
~
and b~ is 60o
ans =1
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ENGINEERING MATHEMATICS 4 VECTORBA501
Vector in a Cartesian plane
Example
Fig 3
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In a Cartesian plane, if point C is (x,y), hence position vector for C can
expressed in the form ;
C(x, y)
xi
yj
xO
OC = Xi + Y j or (Y
X) .
Magnitude for OC , OC = 22+ yx
Unit vector in the positive direction of OC = 22+
+
yx
yjxi
N
A(8, 6)
8
6
y
xO
From fig 3;
ON = 8 , NA = 6,
Hence ON = 8i and NA = 6 j
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ENGINEERING MATHEMATICS 4 VECTORBA501
Operational Vector in Cartesian Plane
Fig 4
Addition and Subtraction of position vectors
Given OP = 1
~r = X1 i + Y1 j or (
1
1
Y
X
) ;
OQ = 2
~r = X2 i + Y2 j or (
2
2
Y
X
)
Hence 1
~r + 2
~r = X1 i + Y1 j + X2 i + Y2 j
= (X1 + X2) i + (Y1 + Y2) j ( assemble i and j )
1
~r – 2
~r = X1 i + Y1 j - X2 i - Y2 j
= (X1 - X2) i + (Y1 - Y2) j
Orin column vector,
1
~r + 2
~r = (
1
1
Y
X
) + (2
2
Y
X
) =
+
+
21
21
Y Y
X X
1
~r – 2
~r = (
1
1
Y
X
) - (2
2
Y
X
) =
−−
21
21
Y Y
X X
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Q(X2, Y
2)
1
~r
y
xO
P(X1, Y
1)
2
~r
In fig 4, P(X1,Y
1) and Q(X
2,Y
2) are
two point in Cartesian Plane
and O is origin.
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ENGINEERING MATHEMATICS 4 VECTORBA501
Example
6 : Given a~ = 5i + 2 j and b
~ = 2i – 5 j , find;
a) a~ + b
~ b) a~ - 2 b
~
Then find magnitude for each vector.
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ENGINEERING MATHEMATICS 4 VECTORBA501
7: Given that position vector A is a~= 2i + 3 j and Position vector B is
b~= i – 5 j . Find :
a) angle between vector AB and A
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Exercise: 1. Given 2 vector a~ = 2i + 3 j and b~ = i + 2 j . If a~ perpendicular to a
~+ λb~ ,Find th
scalar value of λ (ans, : λ =813− )
2.Given a~ =
−4
3and b
~=
−
1
1
a) Find a~+ b
~ (
−3
2)
b) Calculate |a~ | ( 5 unit)
c) If a~= cb ~2~
3 + , Express c~ as column vector (
−
2
13
3
)
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y
r S
y
x
γ
βS
α
ENGINEERING MATHEMATICS 4 VECTORBA501
Vector in Three Dimension in Cartesian Plane
59
O
z
x
p[
O
zDirection Cosines
If OS makes angle of α, β and γ with the coordinat
axes i, j, and k respectively, then
The direction cosines for OS vector are:
cos α, = OS
x
, cos β = OS
y
, cos γ = OS
z
{Magnitude OS = a~ = 222 z y x ++ }
α = angle between vector S and x - axes
β = angle between vector S and y - axes
γ = angle between vector S and z - axes
α, β and γ known as direction angle.
Coordinate S (p, q, r).
Position vector for S.
OS = a~ = xi + yj + zk =
z
y
x
= pi + q j + r k =
r
q
p
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ENGINEERING MATHEMATICS 4 VECTORBA501
Example
8: If position vector of point A, B and C are a~= 2i + j + 2k, b~ = 4i + 5j + 3k and
c~ = i - 3j + 2k respectively, find
a) vector AB
b) direction cosines of AB
c) unit vector in direction of a~+ b~+ c~
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ENGINEERING MATHEMATICS 4 VECTORBA501
The Scalar Product of 3 Dimension
(produce scalar value)
Let A = a1i + b1 j + c1k
and B = a2i + b2 j + c2k
B A • = (a1i + b1 j + c1k ) • (a2i + b2 j + c2k )
= a1a2i ⋅i +a1b2i ⋅ j +a1c2i ⋅k + b1a2 j ⋅i + b1b2 j ⋅ j + b1c2 j ⋅k + c1a2k ⋅i
+ c1b2k ⋅ j + c1c2k
⋅k
However , i ⋅i = i ⋅i kos 00 = 1
⇒ i ⋅I = j ⋅ j = k ⋅k = 1
and, i ⋅ j = i ⋅ j kos 900 = 0
⇒ i ⋅ j = j ⋅k = k ⋅I = 0
∴
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B A • = a1a2 + b1b2 + c1c2
Angle between vector A and B
cos θ = B A
B A~~
~~•
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ENGINEERING MATHEMATICS 4 VECTORBA501
The Vector Product of 3 Dimension
(produce vector value)
the vector product can be written in determinant forms as :
Let, A = a1i + b1 j + c1k
and B = a2i + b2 j + c2k
B A× =222
111
cba
cba
k ji
Unit vector perpendicular to B A×
u = B x A
B x A~~
~~
Area of parallelogram to vector A and B where A and B are side by side
63
B A× = k baba jcacaicbcb )()()( 122112211221 −+−−−
Angle between vector A and B
Sin θ = B A
B x A
~~
~~
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A
B
ENGINEERING MATHEMATICS 4 VECTORBA501
Area of parallelogram, B A× = B A~~
sin θ
Volume of Parallelepiped
a parallelepiped is a three-dimensional figure formed by six parallelograms
An alternative method defines the vectors a = (a1, a2, a3), b = (b1, b2, b3) and
c = (c 1, c 2, c 3) to represent three edges that meet at one vertex. The volume
of the parallelepiped then equals the absolute value of the scalar triple
product a · (b × c ):
Volume of Parallelepiped is:
64
t
Ө
|)(| cbaV ו= = |)(| acb ו = |)(| bac ו
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ENGINEERING MATHEMATICS 4 VECTORBA501
Example10 : Find the angle between vector a~and b
~usinga) scalar productb) vector product
Given :a~= 2i + 3 j +k b~= i – 2 j – 6k
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ENGINEERING MATHEMATICS 4 VECTORBA501
11: Unit vector perpendicular to P and Q respectively are 3i – 2 j + 4k and2i + 3 j – k
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ENGINEERING MATHEMATICS 4 VECTORBA501
12. Find the volume of the parallelepiped with adjacent edges PQ , PR and
PS where P (3, 0, 1), Q(−1, 2, 5), R (5, 1,−1), S(0, 4, 2)
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ENGINEERING MATHEMATICS 4 VECTORBA501
Exercise : 1. Angle between two vector a~= i + λ j +2k and b~= 2i + 3 j + k is cos-1
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1. Find the value of λ. (ans λ = -1)
2. If A~ = 5i – 2 j + 3k . B~ = 3i + j – 2k andC ~
= i – 3j +4k . Find
a) A~
• ( B~
x C ~
) (ans-12)
b) A~
x ( B~
x C ~
) (ans 62i + 44 j -74k )
3. If p = 4i – 3 j + 5k . and q = 3i – 5 j – 2k . Find :
i. q p 32 ×
ii. )4(2 pq p +•