Chapter 1 Notes: Units, Physical Quantities, and Vectors (SI Units, Uncertainty, and Vector Math)

1. Learning Goals

• Three fundamental quantities in physics and their SI units, plus how to use prefixes for larger/smaller units
• How to track significant figures and uncertainties in measurements
• Scalars vs. vectors; how to add and subtract vectors graphically
• Vector components and how to use them in calculations
• Unit vectors and how to describe vectors with components using unit vectors
• Two ways to multiply vectors: scalar (dot) product and vector (cross) product

1.1 The Nature of Physics

• Physics is an experimental science: it observes phenomena and seeks patterns
• Physical theories are explanations based on observation and fundamental principles; very well established ideas become physical laws/principles
• Important note on the meaning of "theory": a theory explains observed phenomena using established principles; it is not a guess or speculation
• The development of a physical theory requires asking appropriate questions, designing experiments, and drawing conclusions from results
• The history of the field often involves indirect progress, failed theories, and refinements (e.g., Galileo’s insights followed by Newton’s laws)
• A theory’s validity has a range; it can be disproven but not proven with absolute finality
• Example: Galileo’s feather vs. cannonball illustrates a range of validity due to air resistance; in a vacuum they fall at the same rate
• The concept of extending the range of a principle over time (e.g., Newton’s laws extending Galileo’s insights)

1.2 Solving Physics Problems

• Mastery in physics comes from applying concepts to problems, not just understanding them
• Problem-Solving Strategies are provided in each chapter to set up and solve problems efficiently and accurately
• Steps are followed by worked Examples to demonstrate techniques
• Bridging Problems at chapter ends combine multiple key ideas
• Four stages of problem solving (I SEE):
  • IDENTIFY the relevant concepts, target variables, and known quantities
  • SET UP the problem: choose the equations; draw sketches; estimate expected results
  • EXECUTE the solution: perform the calculations
  • EVALUATE the answer: compare with estimates; consider limiting cases
• Idealized models simplify complex systems to make problems tractable while preserving essential features
  • Example: modeling a baseball as a point particle, neglecting air resistance to analyze motion; real systems require awareness of the model’s validity

1.3 Standards and Units

• Physical quantities require standard units; a measurement has both a number and a unit
• Operational definitions define units by measurement procedures (e.g., distance by ruler, time by stopwatch)
• A standard defines a fixed unit to enable reproducible measurements
• SI base units: length (meter, m), mass (kilogram, kg), time (second, s)
• Time standard: second defined via cesium-133 atomic transitions: 1 s = 9,192,631,770 cycles of microwaves between two hyperfine states
• Length standard (meter): historically linked to krypton-86 emission length; current definition ties to the speed of light: the meter is defined so that light travels exactly 299,792,458 meters in 1 second; effectively, 1 m is the distance light travels in vacuum in 1/299,792,458 s
• Mass standard: kilogram defined by a platinum–iridium alloy cylinder kept at BIPM; a true atomic standard would be more fundamental but macroscopic realization is used today
• Prefixes (SI): kilo- (10^3), milli- (10^-3), micro- (10^-6), nano- (10^-9), etc.; larger units scale by factors of 10^3; e.g., 1 km = 10^3 m, 1 cm = 10^-2 m
• British (non-SI) units exist but SI units are used in problems; approximate British equivalents may be given for familiarity; examples include 1 inch = 2.54 cm, 1 newton = 1 kg·m/s^2, 1 lb ≈ 4.44822 N
• The International Kilogram in SI and unit compatibility: always ensure consistency of units in calculations

1.4 Unit Consistency and Conversions

• Equations must be dimensionally consistent: each term in an equation must have the same units
• In calculations, treat units as algebraic factors (multiplication/division) to perform unit conversions
• Unit multipliers allow conversion between units without changing the physical quantity
• Typical approach: prefer SI base units (meters, kilograms, seconds) and convert to other units only at the end if needed
• Example: Converting speed from mi/h to m/s
  • Given: 1 mi = 1.609 km, 1 km = 1000 m, 1 h = 3600 s
  • 763 mi/h = 763 × (1.609 × 10^3 m) / (3600 s) ≈ 341 m/s
  • Heuristic: m/s value is a bit less than mph value divided by 2; and about one third of km/h value
• Example: Converting volume 1.84 in^3 to cm^3 and m^3
  • 1 in = 2.54 cm
  • 1 in^3 = (2.54 cm)^3 = 16.387 cm^3; 1.84 in^3 ≈ 30.2 cm^3 ≈ 3.02 × 10^-5 m^3
• Unit consistency tools: common conversions are listed in Appendix E and related sections; always verify units cancel appropriately
• Practice problems illustrate converting between customary and metric units

1.5 Uncertainty and Significant Figures

• Measurements have uncertainties (errors); the uncertainty indicates the maximum likely difference from the true value
• Notation: a measurement such as 56.45(4) mm means 56.45 mm with uncertainty in the last digits; the (4) indicates the uncertainty in the final digits
• Fractional error (or percent error) is the ratio of the uncertainty to the measured value; small percent errors can be significant
• Significant figures rules:
  • When multiplying/dividing, the result has no more significant figures than the factor with the fewest significant figures
  • When adding/subtracting, the result has the same number of decimal places as the value with the fewest decimal places
• Guidelines for reporting: typical instructional problems present numbers with about 3 significant figures; round final answers to the correct number of significant figures; do not truncate, but round
• Example: compute energy using E = mc^2; mass m given to three significant figures; final answer must be rounded to three significant figures; discuss order of magnitude vs. precision
• Scientific notation is useful for very large/small numbers and clearly conveys significant figures (e.g., 3.84 × 10^8 m is three-significant-figure precision)
• Distinguish precision (repeatability) vs. accuracy (closeness to true value)

1.6 Estimates and Orders of Magnitude

• Order-of-magnitude estimates (back-of-the-envelope) are valuable for gaining intuition or checking plausibility
• Not always precise; can be off by factors of 2, 10, or more
• Enrico Fermi popularized these kinds of estimations
• Problems in this section guide rough estimates (e.g., mass/volume estimates for large numbers, or rough counts like teeth on a campus)

1.7 Vectors and Vector Addition

• Scalars are quantities described by a single number; vectors have both magnitude and direction
• Displacement is a vector; it is the straight-line change in position, directed from start to end, regardless of the actual path taken
• Notation: vectors are printed in boldface with an arrow, e.g., you should write
  • The magnitude of a vector is a scalar and is always nonnegative
• Equality of vectors: two vectors are equal only if their magnitudes and directions are identical; vectors can be placed at different points yet be equal
• Negative of a vector: the same magnitude but opposite direction; A = 87 m south → -A = 87 m north
• Antiparallel: vectors with opposite directions can have opposite sense (antiparallel) but may differ in magnitude
• Parallel and antiparallel vectors illustrate how vector magnitudes combine differently from scalar magnitudes
• Vector addition (three ways to visualize):
  • Place the tail of the second vector at the head of the first (head-to-tail method)
  • Form the parallelogram with the two vectors as adjacent sides and draw the diagonal as the resultant
  • The commutative and associative properties hold for vector addition
• Vector sum magnitude cannot, in general, be obtained by simply adding magnitudes: |A + B| ≠ |A| + |B| in general; only when A and B are parallel (same direction) do we have |A + B| = |A| + |B|
• Special cases:
  • If A and B are antiparallel, |A + B| = ||A| - |B||
• For more than two vectors, sum sequentially or use the parallelogram constructions; the order of addition is irrelevant

1.8 Components of Vectors

• Cartesian components: any vector A in the xy-plane can be written as A = Ax hat{i} + Ay
  hat{j}
• Components Ax and Ay are scalars; they describe the projection of A onto the x- and y-axes
• If A has magnitude |A| = A and direction angle θ measured from the +x axis toward +y, then:
  • A_x = A cos θ
  • A_y = A sin θ
• The magnitude and direction from its components:
  • |A| = sqrt(Ax^2 + Ay^2)
  • θ = arctan(Ay / Ax) (with quadrant corrections)
• Caution: angle definitions depend on the chosen reference axis; if angles are defined differently, the expressions change accordingly
• In three dimensions, include A_z and the z-axis with hat{k}

1.9 Unit Vectors

• Unit vectors are vectors with magnitude 1 used to describe direction
• In Cartesian coordinates, the three standard unit vectors are:
  • \hat{i} along +x, \hat{j} along +y, \hat{k} along +z
• Any vector can be written in unit-vector form:
  $\mathbf{A} = A<em>x \,\hat{i} + A</em>y \,\hat{j} + A_z \,\hat{k}$
• The relationships between a vector and its components can be written as:
  $A<em>y = A \sin\theta, \quad A</em>x = A \cos\theta, \quad A<em>z = A</em>z$
• Unit vectors are not vectors themselves in the sense that their magnitudes are fixed at 1; their purpose is to indicate direction
• The unit-vector representation is especially convenient for vector sums in 3D:
  $\mathbf{R} = R<em>x \,\hat{i} + R</em>y \,\hat{j} + R_z \,\hat{k}$

1.10 Products of Vectors

• Scalar (dot) product: A · B is a scalar
  • Geometric definition: $\mathbf{A} \cdot \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \cos\phi$ where φ is the angle between A and B
  • Component form: $\mathbf{A} \cdot \mathbf{B} = A<em>x B</em>x + A<em>y B</em>y + A<em>z B</em>z$
  • Properties: commutative, i.e., $\mathbf{A} \cdot \mathbf{B} = \mathbf{B} \cdot \mathbf{A}$
  • If A ⟂ B, then $\mathbf{A} \cdot \mathbf{B} = 0$
  • Angle from the dot product: $\phi = \cos^{-1}\left( \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{A}| |\mathbf{B}|} \right)$
• Vector (cross) product: A × B is a vector
  • Magnitude: $|\mathbf{A} \times \mathbf{B}| = |\mathbf{A}| |\mathbf{B}| \sin\phi$
  • Direction: perpendicular to the plane containing A and B; given by the right-hand rule
  • Not commutative: A × B ≠ B × A; in fact, A × B = - (B × A)
  • Components of the vector product can be written as:
    $\mathbf{A} \times \mathbf{B} = (A<em>y B</em>z - A<em>z B</em>y) \,\hat{i} + (A<em>z B</em>x - A<em>x B</em>z) \,\hat{j} + (A<em>x B</em>y - A<em>y B</em>x) \,\hat{k}$
  • In terms of components, the scalar triple checks include:
    $(\mathbf{A} \times \mathbf{B}) \cdot \mathbf{C} = \det\begin{vmatrix} \hat{i} &amp; \hat{j} &amp; \hat{k} \ A<em>x & A</em>y &amp; A<em>z \ B</em>x &amp; B<em>y & B</em>z \end{vmatrix} \cdot \mathbf{C}$
• Right-handed coordinate systems: standard practice is to use right-handed systems; left-handed systems are discouraged
• Determinant form for cross products is available but not required for all problems
• Examples illustrate using dot and cross products to solve problems and relate to projections, magnitudes, and angles

1.11–1.16 Vector operations and conventions (summary)

• Adding two vectors: several constructions show that addition is associative and commutative; the resultant R = A + B + C can be found in multiple orders
• Subtracting vectors: defined via addition with a negative vector: A − B = A + (−B)
• Scaling a vector by a scalar c: cA scales magnitude by |c| and reverses direction if c < 0
• Vector addition in right-angled cases; use Pythagoras and trigonometry to resolve components and compute magnitudes and directions
• Dot and cross products can be used to analyze work (dot product with force and displacement), magnetic forces, torque, angular momentum, and more
• Example problems illustrate computing a resultant, analyzing components, and verifying results with sketches

1.17–1.21 Practical vector operations (highlights)

• Using components to verify vector sums; the x- and y-components add independently:
  $R<em>x = \sum A</em>{xi}, \quad R<em>y = \sum A</em>{yi}, \quad R<em>z = \sum A</em>{zi}$
• Magnitude and direction from components:
  $R = \sqrt{R<em>x^2 + R</em>y^2 + R<em>z^2}, \quad \theta = \tan^{-1}\left(\frac{R</em>y}{R_x}\right)$ (with quadrant checks)
• Unit vectors provide a compact representation: A = Ax \hat{i} + Ay \hat{j} + A_z \hat{k}; and A = |A| (cosines with respect to axes)

1.9–1.12 The Vector Product (Cross Product) in detail

• Cross product magnitude and direction; right-hand rule to determine direction of A × B
• When two vectors are parallel (or antiparallel), A × B = 0
• Components of the cross product in terms of A and B components (1.27):
  Cx = Ay Bz - Az By, Cy = Az Bx - Ax Bz,
  Cz = Ax By - Ay B_x
• Cross product can also be written using unit vectors or determinants; right-handed coordinate system is assumed
• Example: calculating A × B using both the geometric rule and components to verify results

1.23–1.25 Additional vector identities and problems

• Practice problems illustrate using unit vectors, vector components, dot and cross products to find angles, magnitudes, and directions
• Distinguish when cross product direction changes with orientation of A and B
• Recognize that scalar and vector products have different geometric meanings and magnitudes; they respond differently to angle changes

Quick reference: Key formulas from Chapter 1

• Dot product (scalar product):
  $\mathbf{A} \cdot \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \cos\phi = A<em>x B</em>x + A<em>y B</em>y + A<em>z B</em>z$
• Angle from dot product:
  $\phi = \cos^{-1}\left( \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{A}| |\mathbf{B}|} \right)$
• Cross product (vector product):
  $\mathbf{A} \times \mathbf{B} = (A<em>y B</em>z - A<em>z B</em>y) \,\hat{i} + (A<em>z B</em>x - A<em>x B</em>z) \,\hat{j} + (A<em>x B</em>y - A<em>y B</em>x) \,\hat{k}$
  Magnitude: $|\mathbf{A} \times \mathbf{B}| = |\mathbf{A}| |\mathbf{B}| \sin\phi$
• Vector sum (displacement addition):
  \mathbf{R} = \mathbf{A} + \mathbf{B} \
  \text{for components: } Rx = Ax + Bx, \; Ry = Ay + By, \; Rz = Az + B_z
• Magnitude from components:
  $|\mathbf{R}| = \sqrt{R<em>x^2 + R</em>y^2 + R_z^2}$
• Direction from components (example): if vector lies in the plane,
  $\theta = \tan^{-1}\left( \frac{R<em>y}{R</em>x} \right)$ (with quadrant checks)

Connecting to earlier and later topics

• Vectors will be essential throughout physics for quantities with direction: velocity, force, displacement, momentum, etc.
• The component method generalizes to 3D and to other vector-like quantities (e.g., forces) studied later in the course
• The practice of dimensional analysis and unit checks (1.4) carries through to all subsequent chapters

Quick practice references

• Example problems referenced: vector addition at right angles (1.5), addition with components (1.7, 1.8), vector products in 2D and 3D (1.11–1.12, 1.25–1.29), and using unit vectors (1.9, 1.23–1.24)
• Test Your Understanding and Bridging Problems appear throughout Chapter 1 to reinforce concepts