Chapter 1 Outline
Units, Physical Quantities, and Vectors
• Idealized models
• Units
• SI units, prefixes, and unit consistency
• Uncertainty and significant figures
• Order of magnitude approximations
• Vectors and scalars
• Component notation
• Vector addition and subtraction
• Dot and cross products
Physics
• Physics is an experimental science.
• Observation leads to theory.
• Theories have limits, or ranges of
validity.
• Solving problems – Idealized
models
• We make approximations in order to
solve problems; you wouldn’t use
general relativity to solve the problem
of a body in free-fall!
• Always keep in mind what
simplifications are inherent in your
model, and think about whether they
are reasonable.
Standards and Units
• We need to describe most physical phenomena
qualitatively, so we compare our measurement of a
physical quantity to some standard reference, or unit.
• The standard system of units is the International System,
or SI.
• In the United States, we often use US (or British) customary units
(inches, pounds…), but we will only use SI units in this course
• The SI base units (Other units are derived from these.)
• Length: meter (m)
• Mass: kilogram (kg)
• Time: second (s)
• Electric current: ampere (A)
• Thermodynamic temperature:
kelvin (K)
• Amount of substance: mole (mol)
• Luminous intensity: candela (cd)
Unit Prefixes
• In order to introduce larger or smaller units, we use
prefixes.
• These are some of the more common prefixes:
Power of Ten
Prefix
Abbreviation
1012
tera-
T
109
giga-
G
106
mega-
M
103
kilo-
k
10−2
centi-
C
10−3
milli-
m
10−6
micro-
μ
10−9
nano-
n
10−12
pico-
p
10−15
femto-
f
Unit Consistency
• All equations must be dimensionally consistent.
• Each side of the equation (or any terms that are added) must have
the same units
• For example: A body moving at a constant speed 𝑣 =
2 m/s will travel a distance 𝑑 = 10 m in time 𝑡 = 5 s.
𝑑 = 𝑣𝑡
m
m
10 m = 2
5 s = 10 s = 10 m
s
s
• Likewise, for unit conversions, you multiply by terms that
60 s
are equal to one, such as
.
1 min
mi
60 mi/hr = 60
hr
1609 m
1 mi
1 hr
= 26.82 m/s
3600 s
Uncertainty and Significant Figures
• All measurements have some level of uncertainty.
• We can express this uncertainty (or error) as a number plus/minus
the uncertainty.
• For example, if the mass of a steel ball is given as 1.24 ± 0.03 kg,
then the ball is unlikely to be greater than 1.27 kg or less than
1.21 kg.
• This uncertainty can also be expressed as a fraction or percent of
the given value.
• If the uncertainty is not explicitly stated, we can go by the
number of significant figures (s.f.).
• We assume an uncertainty of one in the least significant digit.
• When multiplying and dividing, the answer has the same number
of s.f. as the term with the fewest s.f.
• When adding, use the location of the decimal point.
Order of Magnitude Estimates
• It is important to develop a sense of what is reasonable to
expect for an answer to a question.
• If you are calculating the speed of a pitched baseball, would
4 m/s be reasonable? 40 m/s? 400 m/s?
• Sometimes, you can catch errors in your solution by examining
the plausibility of your calculated answer.
• Also, there are times when the data needed to do exact
calculations are not available.
• In this case, we might make an order of magnitude estimate.
Order of Magnitude Estimates Example
• Problem 1.21 – How many times does a typical person
blink their eyes in a lifetime?
Vectors and Scalars
• Some physical quantities are fully described by a single
number with a unit, such as mass, length, time.
• These are scalar quantities, and only have a magnitude.
• We can use regular arithmetic to combine these quantities.
• Other physical quantities, such as velocity or force, must
include a direction.
• These are vector quantities, and have both a magnitude and a
direction.
• We must use vector arithmetic to combine these quantities.
Vector Notation
• First, some notation issues:
• Vectors are drawn as an arrow pointing in the vector’s direction
and a length proportional to its magnitude.
• Multiplying a vector by −1 reverses the direction of the vector.
• Vectors are represented by a letter, generally in bold, with an
arrow, such as 𝑨.
• Unit vectors have a magnitude of one, and are therefore used to
show direction. They are distinguished by a caret or “hat” instead
of an arrow, e.g., 𝒙.
• The magnitude of a vector is a scalar quantity and is normally
written as simply the letter without the arrow. It is also sometimes
written as the absolute value of the vector.
(Magnitude of 𝑨) = 𝑨 = 𝐴
Displacement
• One of the simplest vectors is
displacement, the change in
position of an object.
• Consider the case of walking from the
library to the ISA building.
• Your displacement is approximately
400 m to the northwest.
• The distance you walk depends on the
path you took. Maybe you stopped at
the Marshall center, or maybe you
took the most direct route, annoying
drivers as you cut diagonally across
the roads.
Vector Addition
• Vector addition: 𝑪 = 𝑨 + 𝑩
• Add vectors “head to tail.”
• The order doesn’t matter.
𝑪=𝑨+𝑩= 𝑩+𝑨
• When adding parallel vectors,
the resulting magnitude is the
sum of the two vector
magnitudes.
• When adding antiparallel
vectors, the resulting magnitude
is the difference of the two
vector magnitudes.
• When adding more than two
vectors, they can be grouped in
any combination and order.
Vector Subtraction
• Vector subtraction: 𝑫 = 𝑨 − 𝑩
• Subtracting 𝑩 is the same as adding a negative 𝑩.
𝑫 = 𝑨 + −𝑩
• The order does matter.
𝑫 = 𝑨 − 𝑩;
𝑫 = −𝑬
𝑬=𝑩−𝑨
Multiplication of a Vector by a Scalar
• Multiplication by a scalar:
𝑮 = 𝑐𝑨
• Resulting magnitude is the
product of 𝑐 and 𝐴
• Multiplying a unit vector
𝒙 by a scalar 𝐺𝑥 gives a
vector along the 𝑥
direction with magnitude
𝐺𝑥 .
• This is the basis of the
component form of vectors.
Vector Components
• If we set up an orthogonal
coordinate system, we can
express any vector in terms of
its components along each of
the axes.
• We can use trigonometry to find
the magnitudes of the
components. For a right triangle:
• sin 𝜃 =
opposite
hypotenuse
• cos 𝜃 =
adjacent
hypotenuse
• tan 𝜃 =
opposite
adjacent
Unit Vectors
• Unit vectors have a magnitude of one,
and are distinguished by a caret or “hat.”
• In a coordinate system, unit vectors point
along the positive axes.
• In a Cartesian coordinate system, there
are three axes, 𝑥, 𝑦, and 𝑧.
• The corresponding unit vectors are 𝒙, 𝒚, and
𝒛, or equivalently 𝒊, 𝒋, and 𝒌.
• In component form, subscripts
distinguish the component of the vector
along each axis.
𝑨 = 𝐴𝑥 𝒙 + 𝐴𝑦 𝒚 + 𝐴𝑧 𝒛
or,
𝑨 = 𝐴𝑥 𝒊 + 𝐴𝑦 𝒋 + 𝐴𝑧 𝒌
Vector Components
• Measuring the angle 𝜃
counterclockwise from the 𝑥 axis,
we can find the components from
the magnitude and direction.
𝐴𝑥 = 𝐴 cos 𝜃
𝐴𝑦 = 𝐴 sin 𝜃
• Likewise, from the components,
we can find the magnitude and
direction.
𝐴=
𝜃=
𝐴2𝑥 + 𝐴2𝑦
tan−1
𝐴𝑦
𝐴𝑥
There is always some ambiguity when using the inverse tangent function.
Any two angles that differ by 180° will have the same tangent. By drawing
the vector, it will be apparent which angle to use.
Vector Addition by Components
• Once vectors are expressed in terms of their components,
addition and subtraction is trivial.
• The 𝑥, 𝑦, and 𝑧 components are each added/subtracted amongst
themselves, without mixing.
𝑨 = 𝐴𝑥 𝒙 + 𝐴𝑦 𝒚 + 𝐴𝑧 𝒛
𝑩 = 𝐵𝑥 𝒙 + 𝐵𝑦 𝒚 + 𝐵𝑧 𝒛
𝑪=𝑨+𝑩
𝑪 = (𝐴𝑥 + 𝐵𝑥 )𝒙 + (𝐴𝑦 + 𝐵𝑦 )𝒚 + (𝐴𝑧 + 𝐵𝑧 )𝒛
𝑫=𝑨−𝑩
𝑫 = (𝐴𝑥 − 𝐵𝑥 )𝒙 + (𝐴𝑦 − 𝐵𝑦 )𝒚 + (𝐴𝑧 − 𝐵𝑧 )𝒛
Scalar (Dot) Product
• Many physical relationships can be
expressed by the product of vectors,
but we cannot use ordinary
multiplication with vectors.
• Multiplying vectors using the dot product
results in a scalar quantity.
• The dot, or scalar product can be
calculated using the magnitudes of the
vectors and the angle between them.
𝑨 ∙ 𝑩 = 𝐴𝐵 cos 𝜙
• From this definition, it is clear that the
dot product is at a maximum when 𝑨
and 𝑩 are parallel (𝜙 = 0°) and that
the dot product is zero when 𝑨 and 𝑩
are perpendicular.
Scalar Product by Components
• To find the dot product by components, we look at the dot
products of the unit vectors.
• Since the dot product of perpendicular vectors is zero,
𝒙∙𝒚=𝒚∙𝒛=𝒛∙𝒙=0
• But, the dot product of parallel vectors is the product of their
magnitudes,
𝒙∙𝒙=𝒚∙𝒚=𝒛∙𝒛=1
• Multiplying this all out,
𝑨 ∙ 𝑩 = (𝐴𝑥 𝒙 + 𝐴𝑦 𝒚 + 𝐴𝑧 𝒛) ∙ (𝐵𝑥 𝒙 + 𝐵𝑦 𝒚 + 𝐵𝑧 𝒛)
𝑨 ∙ 𝑩 = 𝐴𝑥 𝐵𝑥 + 𝐴𝑦 𝐵𝑦 + 𝐴𝑧 𝐵𝑧
• We readily see that the dot product is commutative.
𝑨∙𝑩=𝑩∙𝑨
Vector (Cross) Product
• Multiplying vectors using the cross
product results in a vector.
• 𝑪=𝑨×𝑩
• The resulting vector is perpendicular
to both vectors 𝑨 and 𝑩.
• The direction is given by the right
hand rule. Point the index finger of
your right hand along 𝑨 and curl your
fingers towards 𝑩. Your thumb is
pointing in the direction of 𝑨 × 𝑩.
• The magnitude of the cross product
is.
𝑨 × 𝑩 = 𝐴𝐵 sin 𝜙
• The cross product is zero when 𝑨
and 𝑩 are parallel.
Vector Product by Components
• To find the cross product by components, we look at the cross
products of the unit vectors.
• Since the cross product of parallel vectors is zero,
𝒙×𝒙=𝒚×𝒚=𝒛×𝒛=0
• But, using the right hand rule,
𝒙 × 𝒚 = −𝒚 × 𝒙 = 𝒛
𝒚 × 𝒛 = −𝒛 × 𝒚 = 𝒙
𝒛 × 𝒙 = −𝒙 × 𝒛 = 𝒚
• Multiplying this all out,
𝑨 × 𝑩 = (𝐴𝑥 𝒙 + 𝐴𝑦 𝒚 + 𝐴𝑧 𝒛) × (𝐵𝑥 𝒙 + 𝐵𝑦 𝒚 + 𝐵𝑧 𝒛)
𝑨 × 𝑩 = 𝐴𝑦 𝐵𝑧 − 𝐴𝑧 𝐵𝑦 𝒙 + 𝐴𝑧 𝐵𝑥 − 𝐴𝑥 𝐵𝑧 𝒚 + (𝐴𝑥 𝐵𝑦 − 𝐴𝑦 𝐵𝑥 ) 𝒛
• We readily see that the cross product is not commutative.
𝑨 × 𝑩 = −𝑩 × 𝑨
Vector Product by Determinant
• We can also express the cross product in determinant form.
𝒙
𝑨 × 𝑩 = 𝐴𝑥
𝐵𝑥
𝒚
𝐴𝑦
𝐵𝑦
𝒛
𝐴𝑧
𝐵𝑧
𝑨 × 𝑩 = 𝐴𝑦 𝐵𝑧 − 𝐴𝑧 𝐵𝑦 𝒙 + 𝐴𝑧 𝐵𝑥 − 𝐴𝑥 𝐵𝑧 𝒚 + (𝐴𝑥 𝐵𝑦 − 𝐴𝑦 𝐵𝑥 ) 𝒛
Chapter 1 Summary
Units, Physical Quantities, and Vectors
• Idealized models – Know your assumptions
• Units
• SI units, prefixes, and unit consistency
• Uncertainty and significant figures
• Order of magnitude approximations – Is the answer reasonable?
• Vectors (magnitude and direction) and scalars (magnitude)
• Component notation and unit vectors
• Vector addition and subtraction (graphically and by components)
• Dot product: 𝑨 ∙ 𝑩 = 𝐴𝐵 cos 𝜙 = 𝐴𝑥 𝐵𝑥 + 𝐴𝑦 𝐵𝑦 + 𝐴𝑧 𝐵𝑧
• Cross product: 𝑨 × 𝑩 = 𝐴𝐵 sin 𝜙
𝑨 × 𝑩 = 𝐴𝑦 𝐵𝑧 − 𝐴𝑧 𝐵𝑦 𝒙 + 𝐴𝑧 𝐵𝑥 − 𝐴𝑥 𝐵𝑧 𝒚 + (𝐴𝑥 𝐵𝑦 − 𝐴𝑦 𝐵𝑥 ) 𝒛