Post on 20-Jul-2020
28/06/13 Vectors (© F.Robilliard) 1
x
y
s
θ" 4
3
2.8
28/06/13 Vectors (© F.Robilliard) 2
Introduction:
To make precise descriptions of the physical world we define measurable physical quantities that represent particular aspects of that world. These quantities must be realizable experimentally, and are specified as numeric multiples of a standard amount of
the quantity, called a unit.
A model is then constructed, that precisely describes the interrelationships, that exist between the various quantities
associated with a given physical system.
In these notes, we will identify two different types of quantity used in Physics, and develop methods for their use.
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Scalar & Vector Quantities:
We find two types of quantity in Physics:
Scalars: have a magnitude only
Vectors: have a magnitude and act in a direction in space
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Examples: Scalars: mass m time t
volume V energy E density ρ
temperature T
Vectors: displacement r
velocity v acceleration a
force f momentum p
electric field vector E area A
Is area a scalar, or a vector?
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Representation of Scalars:
Scalars can be fully represented by a single number, representing their magnitude in terms of a unit–
Example: mass = 5 kg.
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Representation of Vectors:
The direction of the arrow corresponds to the direction of the vector in space, relative to stated reference axes.
For Vectors, magnitude and direction must both be represented.
This can be done using an arrow, drawn on a graph page.
The length of the arrow is proportional to the magnitude of the vector, according to a stated scale.
Note: Because our vectors exist in 3-dimensional space, they require three numbers for their representation.
Note: the location of the arrow doesn�t matter – only its length, and angle of inclination.
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Example: Represent a force of 3.0 N acting on a body in the +x direction
3.0 N
x
y
x
y Scale:
1 cm = 1 N
2.8 N
x
y
x
y Scale:
1 cm = 1 N
Represent a force of 2.8 N acting on a body at 45 deg to the +x direction
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Notation: The fact that a symbol represents a vector quantity is
indicated in a number of alternative ways:
The magnitude of a vector can be explicitly represented by the use of modulus bars, or by not using the above notational
devices
(Magnitude of vector v) = |v| = v Not bold
v :gleUndersquig v :Underarrow :Font Bold
~
v
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Addition of Vectors:
If two, or more, forces act on the same body, the motion of the body will be determined by the total overall effective
force (also called the resultant force), acting on the body. To find this we add the individual forces.
The sum, s, of two vectors a and b, is represented by s = a + b
Notation:
In many physical situations, we need to know the total effect of several simultaneous individual vectors. For example -
If each of two charges, produces an electric field force at a point in space, the total field force at that point will be the sum of the two individual field force components.
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Rule for Graphical Addition of Vectors:
Arrange the arrows representing the individual vectors, on a graph page so that the head of one touches the tail of the next.
Draw a new arrow from the tail of the first vector, to the head of the last vector.
This new arrow represents the sum of the individual vectors.
Note: the sum will be the same, no matter in what order the arrows are placed.
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Example: Add 2 Vectors Two forces act
on a body
4 N
3 N
4
3 s
Total effective (resultant) force acting on the body is 5 N acting at 36.9 deg to the direction of the 4 N force
θ"
Scale: 1 cm = 1 N
Using a ruler: s has a magnitude of 5
( |s| = 5 units)
Using a protractor: s has a direction of 36.9 deg
above the 4 N force ( θ = 36.9 deg )
s = (4 + 3) N
5 N
36.90
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Example: Add 3 Vectors
Using a ruler: s has a magnitude of 5.4
( |s| = 5.4 units)
Using a protractor: s has a direction
of 68.2 deg to the +x-axis
( θ = 68.2 deg )
4 ) + ( 3 ) + ( 2.8 S = ( )
x
y
s
θ" 4
3
2.8
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Vector Subtraction:
Negative Vector: To get the negative of a vector, reverse its direction
- ( ) = ( 3 3 )
Subtraction: To subtract a vector, add the negative vector
( ) – ( 3 ) = ( 4 4 ) + ( 3 )
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Magnitude of d: | d | = 5 units
Direction of d: θ = 36.9 deg
Example: Difference Vector:
x
y
4
3 d
θ"
( ) ( ) ( ) ( ) ( )3434d vectordifference~
↓+=↑−=≡→→
Subtract a vector, of magnitude 3 units in the +y-direction, from a vector of magnitude 4 units in the +x-direction.
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Resolving Vectors: This is the inverse of addition. Any single vector, v, can be
replaced by two vectors, a and b, which are perpendicular to each other, and which add together to equal v.
v = a + b a b
a = v cos θ b = v sin θ
a and b are the components, or resolved parts, of v along the x & y axes, respectively.
v b
a a
v b
x
y
θ"
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Unit Vectors:
The unit vector pointing in the direction of a vector v, is represented by ) ( v̂ hat" v" pronounced
( )
|v|
v v̂ ~
vector)of (magnitude
(vector)runit vecto
≡
=
These are vectors whose magnitude is one unit.
v̂v
3 ∧
3
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i, j and k unit vectors:
x
y
z
i j
k
These are special unit vectors directed along the x, y, and z axes
Any vector can be expressed in terms of i, j and k unit vectors.
This representation gives us a powerful method for working with vectors.
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Representing Vectors in i, j & k:
v = 4 i + 3 j
v = OP = 4 i + 3 j + 2 k
d = 4 i - 3 j
x
y
4i
3j v
θ"
i j
x
y
4i -3j d
θ"i
j
x
y
z
i
j k
4
3
2
v
P
O
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General i, j & k form:
v = a i + b j = ( v cos θ ) i + ( v sin θ ) j
Resolve a vector v into its components along the x and y axes
a = v cos θ
b = v sin θ a
v b
x
y
θ"
This allows any vector to be expressed in i, j, k form.
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Adding Vectors using i, j &k:
i j
a b
s
a = 2 i + 3 j b = 3 i - 4 j
sum = s = a + b = (2+3) i + (3-4) j = 5 i - j
Graphically: Using i, j, k:
Both methods are equivalent, but i, j, k is preferred.
s = 5 i - j
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Polar & Cartesian forms:
θ" x
y
v
vx = v cos θ
vy = v sin θ"
vx
vy
Polar: magnitude and direction
(v, θ)"
Cartesian: x & y components
v = vx i + vy j
There are two forms in which a vector can be expressed.
Cartesian to Polar: Polar to Cartesian: v2 = vx
2 + vy2 (Pythagoras)
tan θ = vy/vx
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Example: Polar to Cartesian:
A force F of 8 N acts at an angle of 30 deg to the x axis.
Express F in Cartesian form.
x
y F = 8 N
θ = 30o
34
30cos8
cos FFx
=
=
=o
θ
430sin8
sin FFy
=
=
=o
θ
jijiF 4 )3(4 ) sinθ (F ) cosθ (F +=+=
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Example: Cartesian to Polar: Express a force F = (2 i+ 4 j) N, in polar form.
Magnitude:
)(47.45220
2042 22
222
NF
FFF yx
===
=
+=
+=
Direction:
o
x
y
FF
4.63
224tan
=
===
θ
θ
y
2 i
4 j
x
F F = (2 i+ 4 j) N
θ
Answer: F has magnitude 4.47 N and acts at
an angle of 63.4 deg to the +x axis.
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Vector Products: There are two contexts in Physics where it is useful
to define a product of one vector with another. Thus there are two different products defined.
Dot (or Scalar) Product:
This produces a scalar result, and is represented by
a .b
An example of its use is in the definition of work.
Cross (or Vector) Product:
This produces a vector result, and is represented by
a x b
An example of its use is in the definition of magnetic
field vector.
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Dot Product:
Definition: a . b = a b cos θ θ a
b
(a . b) is a scalar equal to the product of the magnitudes of vectors a & b and the cosine of the angle between them.
Alternatively: (a . b) is the product of the magnitude of the component of one vector, in the direction of the other,
with the magnitude of the other. a
b a cos θ
a sin θ θ a.b = (a cos θ).b
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Dot Products between i, j & k:
i . i = 1.1. cos 0o = 1
From definition of the dot product:
i . j = 1. 1 cos 90o = 0
a . b = a b cos θ
The same: i . i = j . j = k . k = 1
Different: i . j = i . k = j . k =0 etc.
Summary:
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Example (a.b):
Find (a . b) where a = 2 i + j + 3 k and b = i - 4j + 3 k
Solution: a . b = (2 i + j + 3 k) . (i - 4 j + 3k) = (2i).(i - 4 j + k) + (j).(i - 4 j + 3k) + (3k).(i - 4 j + 3 k) = 2 - 4 + 9 = 7
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Dot Product General Case:
Say: a = ax i + ay j + az k……….…(1) and b = bx i + by j + bz k………….(2)
(a . b) = (ax i + ay j + az k) . (bx i + by j + bz k) = ax. bx + ay by + az bz ………….(3)
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General Uses of the Dot Product:
1. Test for Perpendicularity: If a . b = 0 & a,b 0 then a b (a & b are “orthogonal”)
Now a . a = ax2 + ay
2 + az
2 [from (1) & (3)] but a . a = a.a cos 0 = a2 hence a2 = ax
2 + ay2
+ az2
2. Find Magnitude of a Vector, a:
)4..(..........( 222zyx aaaaThus ++=m
The dot product has some useful general applications.
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General Uses of the Dot Product:
3. Find Angle Between Two Vectors:
)5(a.b
θcosget We mma.b
=
θ cos a.b :definition From =b . a
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Example: Given: a = 2 i + j + 3 k and b = i - 4j + 3 k Find: the magnitudes of a & b, and the angle, θ, between them. Magnitudes:
14
14914)32).(32(
2
=
=++=
++++=
=
a
akjikji
a.a
26
269161)34).(34(
2
=
=++=
+−+−=
=
b
bkjikji
b.b
θ a
b
Angle Between a & b:
o
ab
ab
5.682614
7cos
cos
=
==∴
=
θ
θ
θa.b
a.b from before
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Cross Product:
The Cross Product (a x b), of two vectors a, & b, is a vector with direction perpendicular to the plane defined by a & b,
and whose magnitude is (a b sin θ)
Definition: θ a
b
^
~cb x a ) ( θsin b a=)(
x
y
z
b
a
(a x b)
θ
θ = angle from a to b (RHR). Rule HandRight by the
given isdirection whoseand & lar toperpendicu
runit vecto a is :where
ba
c^
~
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Right Hand Rule: (a x b) Place the fingers of the right hand, so that they point around
the ends of the two vectors from the first vector, a, to the second vector, b.
The thumb then points in the direction of the cross product (a x b)
Note: order is important (a x b) = -(b x a)
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Cross Products between i, j & k:
i x i = 1.1.sin 0o = 0
^
~cb x a ) ( θsin b a=)( From
i x j = 1.1.sin 90o k = k (using RHR)
Similarly: i x i = j x j = k x k = 0
i x j = k, j x k = i, k x i = j
j x i = -k, k x j= -i, i x k= -j i
j k
To remember:
clockwise order : +(other vector)
anticlockwise order : -(other vector)
x
y
z i
j k
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Example:
Find: (a x b)
Given: a = i + 2 j b = 3 i - j
(a x b) = (i + 2 j) x (3 i - j) =[ i x 3i] + [i x (-j)] + [2j x 3i] + [2j x (-j)] = 0 + (-k) + (-6k) + 0 = -7 k
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Determinates: Evaluating cross products term by term can be tedious. Determinates offer a slick alternative method.
A determinate is a square array which has a value. The rule for determining the value is hierarchical, and based on a definition for the smallest determinate - the 2x2.
2x2 Determinate: (a,b,c & d are numbers)
cbaddcba
−≡
21210
4x32x55432
−=−=
−=
Definition: Example:
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Determinates: 3x3 Determinate: (a,b,…i are numbers) Definition: 3x3 is defined in terms of 2x2’s
Coefficients of the 2x2’s are the top row terms of the 3x3, with alternating signs (+a,-b,+c)
To get the 2x2 after a particular coefficient, mask the row and column that intersect on that coefficient.. Consider the “a”
The 2x2 that remains unmasked, is multiplied by the “a” coefficient
The other (2x2) determinates are found similarly
hged
cigfd
bihfe
aihgfedcba
+−≡
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Example:
63107
)43(3)16(2)18(1
1143
32113
22114
1211143321
−=
−−+=
−+−−−+=
+−+=
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Cross Product by Determinates:
kjibkjia
zy x
zy x
bbb and
aaaGiven
++=
++=
zy x
zy x
bbbaaax:thenkji
b) (a =
Cartesian vectors a and b:
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Example: Given: a = 2 i + 3 j + k
b = i - 4 j + 5 k Find: (a x b)
kjikji
kji
kjiba
11 - 9 - 19 3)-(-8 1)-(10 -4)(15
4-132
5112
54-13
54-1132) x (
=
++=
+−+=
=
(To evaluate cross products, determinates are generally preferred)
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A General Use of the Cross Product:
[ ]b)(abanba
x ) x (ˆ and of plane tonormalr unit vecto =≡
Find the Unit Vector Normal to a Plane:
Say vectors a & b define a plane
Then (a x b) will be perpendicular to both a & b, and consequently to their plane.
a
b
n̂
Unit vector, normal to the plane of a and b, is the unit vector in the direction of (a x b)
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x
y
s
θ" 4
3
2.8