phys201 11314 chapter1 vectoralgebra -...
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http://ctaps.yu.edu.jo/physics/Courses/Phys201/
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Methods of Theoretical Physics 1
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Chapter 1
Vector Analysis
http://ctaps.yu.edu.jo/physics/Courses/Phys201/Chapter1
Vector Algebra Scalars and Vectors
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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Scalars – Scalar Fields
A Scalar is a physical quantity completely
defined by its magnitude.
Examples are: mass, temperature, pressure…
A scalar field is a scalar mathematical function
that defines a certain physical quantity at
every point in a given region of space.
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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A Vector is a mathematical representation of a
physical quantity which should be defined by
its magnitude and direction
Vectors – Vector Fields
Examples are: Force, linear momentum
A vector field is a vector mathematical function
that defines a physical vector at every point in
a given region in space.
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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→A
A vector is represented by an arrow. Its length is
proportional to the length of the quantity it represents.
The arrow points to its direction.
Graphical representation of a vector 1
The vector makes an angle of 45°°°° with the
direction representing the east (E)
→A
The vector represents
the wind speed of
60 km/h whose direction
is North-East.
1 unit = 10 km/h
→A
6 un
its
N
EW
S
The symbol is used to represent a vector.→A
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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A Vector has no position. Thus the vector
also represents the wind speed of 60 km/h whose
direction is North-East.
→B
Graphical representation of a vector 2
→A
6 u
nits
6 u
nits
→B
N
EW
S
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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Length of a vector
We represent the length of a vector by:
| | or simply A.
In the previous example | | = 60 km /h→A
→A
6 u
nits
In this course we shall indifferently use | | or A
to represent the length of a vector
→A
→A
→A
Properties of Vectors
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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The Vector Space
Mathematicians have devised some operations
involving vectors that pertain to physical quantities
that have both magnitude and direction
Mathematicians define a space called the vector
space which has its own “rules” and operations.
As for any space we shall define in the vector space
some properties and mathematical operations such
as addition, subtraction, multiplication, derivation,
etc…
See Supplement 1
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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Properties of vectors1
a) | | = | |, i.e. and have the same magnitude.→A
If and only if:→→
= BA1)
→B
→A
→B
b) // , i.e. and have the same direction.→A
→B
→A
→B
N
EW
S
→A
6 u
nits
6 u
nits
→B
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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Properties of vectors2→→
−= BA If and only if:2)
b) anti// , i.e. and have opposite
directions.
→A
→B
→A
→B
N
EW
S
→A
6 u
nits
6 u
nits
→B
a) | | = | |, i.e. and have the same magnitude.→A
→B
→A
→B
Addition of Vectors
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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We say that vector is the sum (or resultant)
of two vectors and by writing:
Addition of vectors
Two (equivalent) methods are used to add two
vectors:
• The head to tail (or the triangle) method.
• The parallelogram method.
+→A
→B
→C =
→A
→B
→C
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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In this method we draw, starting from the head of
vector (point a) a vector equal to . (b is the head
of the latter).
The head to tail (or the triangle) method
→A
→B
→A
→B
→C
→B
→A
O
a
b
The sum (vector ) is the vector
completing the triangle with b as its head.
→C
→Ob
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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In this method we draw, starting from the tail of
vector (point O) a vector equal to . (b is the
head of the latter). The sum (or vector ) is the
diagonal of the parallelogram formed by the 2
vectors.
The parallelogram method
→A
→A
→B
→C
→B
→Oc
→C
→A
O
a
c
→B
b
→B
→A
Subtraction of Vectors
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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To subtract vector from vector , we use
the definition of
→B
Subtraction of vectors
The previous methods are used to obtain
→A
)B(ABAC→→→→→
−+=−=
→−B
→C
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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The head to tail (or the triangle) method
(or vector ) is the
vector completing the
triangle with b as its
head.
→C
→Ob
In this method we draw,
starting from the head of
vector (point a) a vector
equal to .
→A
→B
O
→A a
→B-
→A
→−B
b
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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In this method we
draw, starting from the
rear of vector (point
O) a vector equal to - .
The parallelogram method
→A
→A
→B
→B
O
→A
→A
a
b
→B
-
c
(or vector ) is the
diagonal of the
parallelogram formed by
the 2 vectors and - .
→C
→Oc
→A
→B
→C
Representation of Vectors
in a Cartesian System of
Coordinates
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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In a right-handed Cartesian system the axes Ox, Oy
and Oz are mutually perpendicular and they verify
the right-hand rule in this order. (conventionally
counter-clockwise)
Right-handed Cartesian System
x
z
y
Back to Components© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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Representing vectors is independent of the
coordinate system we use.
Components of a vector
→A
Projections of a
vector on the axes
of a coordinate
system are called
componentscomponents
x
z
y
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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Components of a vector
→A
Ay
Az
Ax
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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αααα, ββββ and γγγγ are called the angles of direction.
Direction cosines of a vector
→A
α
β
γ
x
z
y
cosαααα, cosββββ and cosγγγγ are called the direction cosines.
Az
Ax
A
AA xx ==αcos| |→A
A
AA yy ==βcos| |→A
A
AA zz ==γcos| |→A
Ay
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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In a right-handed Cartesian system the
components of vector are denoted Ax , Ay and
Az.
Direction Cosines and Components
→A
α
β
γ
x
z
y
Ax = A cos αααα
Ay = A cos ββββ
Az = A cos γγγγ
→A
α = (Ox, )→A
γ = (Oz, )→A
β = (Oy, )→A
Ay
Az
Ax
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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A and the components
→A
Ay
Az
Ax
22
yx AA ++++
O
P
Q
z
y
x
Triangles ORQ, OTP and
OPQ are right angled.
T
R
OR = PQ = Az
22
yx AA ++++RQ = OP =
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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A and the componentsBy Pythagorean theorem, we have:
( ) 21222zyx AAAA ++=
22
222
yx AA ++++====
++++==== TPOTOP
(((( )))) 2222
222
zyx AAAA ++++++++====
++++==== QPOPOQ
OR = PQ = Az
22
yx AA ++++RQ = OP =
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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The previous relations show that if we know the
three components Ax, Ay and Az then we know
exactly the vector itself.
Defining a vector using its components
→A
222zyx AAAA ++=
A
AA yy ==β→
|A|
cos
A
AA xx ==α→
|A|cos
A
AA zz ==γ→
|A|cos
Multiplication in the
Vector Space
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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We define 3 types of multiplication in the
vector space, namely:
Multiplication of a vector by a scalar
• Multiplication of a vector by a scalar.
• The scalar product: here the multiplication
of 2 vectors gives a scalar.
• The vector product: here the multiplication
of 2 vectors gives a (3rd) vector.
Scalar ×××× Vector
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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Multiplication of a vector by a scalar
Given a vector , we define, for any real
number m, the vector:
Obviously we have (m#0):
|||| AC→→
= ma)
→C
→A
m
1=
a) | | = | |→C
→A
m
1
→A
→→= AC m
b)If m > 0 then // ;
If m < 0 then anti //
→A
→C
→A
→C
b)If m > 0 then // ;
If m < 0 then anti //
→A
→C
→A
→C
= m A
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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For a particle of mass m moving with velocity ,
the linear momentum is defined by
a) | | = m | |
Example: Linear momentum
b) // , i.e. the 2 vectors have the same direction.
→→→→p
→→→→v
→p
→v
→p
→v
→→= vmp
→→→→v
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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A charge q when submitted to an electric field
suffers a force given by (Coulomb’s Law)
EqF��
=a)
Another Example: Electric force on a charge
EqF��
====
F�
E�
b) If q > 0 then // ;F�
E�
If q < 0 then anti // F�
E�
F�
E�
E�
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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One can use the previous property to define
the unit vector of any vector. Let’s consider
the vector defined by:
b)
a) a = 1
Unit Vectors
is called the unit vector in the direction of
vector
→a
→A
→→= A1aA
A
→→= Aa
→→A//a
→a
We use the following notation to write this unit vector.
Scalar (Dot) Product
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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We define the scalar product of 2 given vectors
as follows:
Vector times Vector = Scalar
The symbol •••• (dot) is used to indicate that the
product yields a scalar*.
Where symbolizes the angle between the
2 vectors.
( , )^→
A→B
* To distinguish it from the vector product which yields a vector as we said
=| | ×| | cos ( , )→A
→B
→A
→B•
→A
→B
^
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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Scalar product – Magnitude of a vector
We can easily see that
^
=| |2
→→→→=
→•
→A,AcosAAAA | |×| |
→A
The magnitude of vector is the square root of
the scalar product of by itself:
→A
→A
→→→•= AAAA = | |
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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=| | ×| | cos ( , )
The scalar product of 2 perpendicular vectors is
zero.
Orthogonal Vectors
→A
→B
→A
→B•
→A
→B
The 2 vectors are said to be orthogonal.
^
If = 90°°°° then = 0 ( , )^→
A→B
→A
→B•
→A
→B
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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- The scalar product is commutative; i.e.:
This is easily seen from the previous
definition. The reason is simply that
Properties of the scalar product
=→A
→B• •
→B
→A
cos ( , )→A
→B
^= cos ( , )
^→B
→A
- The scalar product is distributive; i.e.:
→→→→→→→•+•=
+• CABACBA
Vector (Cross) Product
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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We define the vector product of 2 given
vectors as follows:
| | =| | ×| | sin( , )→A
→B
^
, and form a right-handed system
Where symbolizes the angle between the 2
vectors.
( , )^
→A
→B
Vector × Vector = Vector
→A
→B
The symbol ×××× (cross) is used to indicate that
the product yields a vector. This product is also
called cross product.
××××××××→A
→B
→C= →
A→B
→C{
→C
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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A right handed system is defined (identified)
using the right-hand rule.
θθθθ
Right-handed Systems
→A
→B
→C
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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A B sinθ is nothing else but the area of the
parallelogram that the 2 vectors and form.
θθθθ
Area of a parallelogram – Area Vector
Thus vector is often called the area
vector of the previous parallelogram
→→→×= BAC
→A
→B
→B
→A
→C
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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≠→→
×AB→→
×BA
This is easily seen from the previous definition.
The reason is simply that
Properties of the vector product
- The vector product is not commutative; i.e.:
sin ( , )→A
→B
^≠ sin ( , )
^→B
→A
- The vector product is distributive; i.e.:
→→→→→→→×+×=
+× CABACBA
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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It can easily be seen that
= -××××××××→A
→B ××××××××
→B
→A
θθθθ
→B
→A
→→→→×−=× ABBA
sin ( , )→A
→B
^= - sin ( , )
^→B
→A
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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Scalar product - ExampleUnder the effect of an external electric field
an electric dipole rotates. The (rotational)
work done is given by:
Similarly under the effect of a magnetic field
a magnetic dipole rotates. The (rotational)
work done is given by:
→B
→→•= EpW
Where is the moment of this dipole.
→→•µ= BW
→E
→p
Where is the moment of the magnetic dipole.→µ
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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The rotation of the dipoles in the previous
examples is due to the torque (moment of the
force) defined by:
in the case of the magnetic dipole.
Vector Product - Example : Torque
in the case of the electric dipole and by:
→→→×=τ Ep
→→→
×µ=τ B
See Phys. 102
Cartesian Systems
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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In a cartesian system of coordinates, the
unit vectors in the directions x, y and z
respectively are called: , and . i j k
Cartesian Systems
z
yi j
k
x
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors
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Defining a Cartesian SystemAlternatively a Cartesian system is defined using
the 3 orthogonal unit vectors.
, and verify the following relations:i j k
z
yi j
k
kji ˆˆˆ =×
ikj ˆˆˆ =×
jik ˆˆˆ =×
0ˆˆˆˆˆˆ =•=•=• ikkjji
1ˆˆˆˆˆˆ =•=•=• kkjjii
These relations define the
orthogonality of the 3 unit
vectors’ system
II
x
Orthonormality
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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2 perpendicular unit vectors are said to be
orthonormalized (orthogonal and normalized)
Orthonormality
k
ji
0ˆˆˆˆˆˆ =•=•=• ikkjji
1ˆˆˆˆˆˆ =•=•=• kkjjii
These relations define the orthonormality of
the 3 unit vectors’ system
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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In a Cartesian system of coordinates, a vector
can be written as follows:
Components of a vector
kji zyxˆAˆAˆAA ++=
→
→A
γ
z
y
Ay
→A
appear as the sum of 3
vectors:
→A
Az
Ax
→→→→++= zyx AAAA
ixxˆAA =
→
jyyˆAA =
→
kzzˆAA =
→ i j
k
x
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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→
zA
→→→→→→→→++++ yx AA
→
zA
→→→→++= zyx AAAA
→A
O
z
y
x
→
xA
→
yA
Scalar Product Using
Components
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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kji zyxˆBˆBˆBB ++=
→
Consider the scalar product of two vectors:
kji zyxˆAˆAˆAA ++=
→
( ) ( ) ( ) 222 ˆBAˆBAˆBABA kji zzyyxx ++=•→→
zzyyxx BABABABA ++=•→→
222x
222x BBBAAA
BABABABAcos
zyzy
zzyyxx
++×++
++=
•=θ
→→
BA
If θθθθ is the angle between the 2 vectors then:
zzyyxx BABABABA ++=•→→
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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Consider the two vectors:
Example 2
ji ˆ3ˆ2A +=→
ji ˆˆ2B +−=→
( ) 11322BA −=×+−×=•→→
°≈−
=
+×+
−=
•=θ
−
−
→→
−
13.9765
1cos
1494
1cos
BAcos
1
11
BA
If θθθθ is the angle between the 2 vectors then:
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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θθθθ
→A
→B
x
y
j
i
°≈θ 13.97
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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Consider the two vectors:
Example 3
kji ˆˆ3ˆ2A −+=→
kji ˆˆˆ2B ++−=→
( ) ( ) 2111322BA −=×−+×+−×=•→→
°=−
=
++×++
−=
•=θ
−
−
→→
−
6.10284
2cos
114194
2cos
BAcos
1
11
BA
If θθθθ is the angle between the 2 vectors then:
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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kji zyxˆBˆBˆBB ++=
→
Consider the vector product of two vectors:
Vector product using components
kji zyxˆAˆAˆAA ++=
→
( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) kkjkik
kjjjij
kijiii
zzyzxz
zyyyxy
zxyxxx
ˆˆBAˆˆBAˆˆBA
ˆˆBAˆˆBAˆˆBA
ˆˆBAˆˆBAˆˆBABA
×+×+×
+×+×+×
+×+×+×=×→→
( ) ( )( )( )( ) ( )( ) ( )( ) 0ˆBAˆBA
ˆBA0ˆBA
ˆBAˆBA0BA
+−+
+++−
+−++=×→→
ij
ik
jk
yzxz
zyxy
zxyx
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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And rearranging we have:
( )( )( )k
j
i
xyyx
zxxz
yzzy
ˆBABA
ˆBABA
ˆBABABA
−
+−
+−=×→→
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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zyx
zyx
kji
BBB
AAA
ˆˆˆ
C =→
The previous result can be obtained using the
definition of determinants of matrices:
Vector product Using Determinants
kjiyx
yx
zx
zx
zy
zy ˆBB
AAˆ
BB
AAˆ
BB
AAC +−=→
Triple Products
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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Definition:
Triple Scalar Product – Mixed Product
→→→ו= CBAd
ו=→→→CBAd
This definition means precisely that:
But the parenthesis are not necessary since
the cross product is not defined→
×Cm
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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Properties of the Mixed Product→→→
ו= CBAd
zyx
zyx
zyx
CCC
BBB
AAA
=
( )( )( )( )
−+−+−
•++=k
j
i
kji
xyyx
zxxz
yzzy
zyx
ˆCBCB
ˆCBCB
ˆCBCBˆAˆAˆA
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
70
1.
We leave the proof of the following properties
as a homework.
- Cyclic permutation relation
→→→ו CBA
zyx
zyx
zyx
CCC
BBB
AAA
→→→ו= BAC
→→→ו= ACB
zyx
zyx
zyx
BBB
AAA
CCC
=
zyx
zyx
zyx
AAA
CCC
BBB
=
2. The following three determinants are strictly equivalent
→→→ו CBA
→→→→→→ו−=ו−= CABBCA
3. Any modification of the order of the 3 vectors
yields a negative sign, i.e.:
HW
2
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
71
θθθθO c
b d
φφφφ→A
- The mixed product represents in fact
the volume of the parallelepiped formed by the
3 vectors.
- Volume of a parallelepiped→→→
ו CBA
Vparallelepiped = B C sinθθθθ A cosφφφφ.
B C sinθθθθ = Area of the base of the
parallelogram (Obdc)
→→×CB
→B
→C
A cosφφφφ = Height of the parallelepiped
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
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Definition:
Triple Vector Product – The BAC-CAB Rule
××=→→→→CBAD
It can easily be seen that:
We leave, as an exercise, the proof of this
relation known as the BAC BAC –– CAB ruleCAB rule.
××=→→→→CBAD
•−
•=
→→→→→→BACCAB
We can also use the properties of the vector
product to predict the result of the triple
vector product.
HW
2
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
73
Geometrical Justification of the BAC-CAB Rule
And is a vector which should lie in
the previous plane and thus it is a linear
combination of the 2 vectors and .→B
→C
××→→→CBA
The cross product is perpendicular to
the plane which contains and .
→→×CB
→B
→C
© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra
74
No general definition of such an operation
exists in the vector space.
Division of a vector by a vector
The only case where such a division is defined
is when the two vectors are parallel (or
antiparallel). That’s why we do not have a
general definition of this operation.