# Lecture 1: Units, Vectors, and Measurement ## Physics I — Mechanics & Thermodynamics --- ## Learning Objectives By the end of this lecture, students will be able to: 1. Identify and convert between SI base units and derived units 2. Perform dimensional analysis to verify equations and solve problems 3. Apply significant figures and error propagation rules correctly 4. Distinguish between scalar and vector quantities 5. Add, subtract, and resolve vectors using both graphical and component methods 6. Compute dot products and cross products and explain their physical meaning --- ## Lecture Content ### I. The SI System of Units Physics is built on quantitative measurement, and every measurement requires a **unit** to have meaning. The International System of Units (SI) defines seven base units from which all other physical units are derived. These base units are the **meter (m)** for length, the **kilogram (kg)** for mass, the **second (s)** for time, the **ampere (A)** for electric current, the **kelvin (K)** for temperature, the **mole (mol)** for amount of substance, and the **candela (cd)** for luminous intensity. For the study of mechanics and thermodynamics, the three most important base units are the meter, kilogram, and second. Derived units are combinations of these base units that describe more complex physical quantities. For example, the Newton is defined as 1 N = 1 kg m/s^2, the Joule as 1 J = 1 kg m^2/s^2, the Pascal as 1 Pa = 1 kg/(m s^2), and the Watt as 1 W = 1 kg m^2/s^3. SI prefixes allow compact notation for very large or very small quantities, ranging from nano (10^-9), micro (10^-6), milli (10^-3), and centi (10^-2) on the small end to kilo (10^3), mega (10^6), and giga (10^9) on the large end. The strategy for unit conversion is to multiply by conversion factors that are equal to 1. For instance, to convert 60 km/h to m/s, one writes 60 km/h x (1000 m / 1 km) x (1 h / 3600 s) = 16.7 m/s. ### II. Dimensional Analysis Every physical equation must be **dimensionally consistent**, meaning the dimensions on both sides must match. Dimensions are denoted with brackets: [L] for length, [M] for mass, and [T] for time. Dimensional analysis serves several purposes: it can check whether an equation is plausible, derive relationships between physical quantities up to a dimensionless constant, and identify errors in algebraic work. As an example, consider whether v = at makes dimensional sense. The left side has dimensions [v] = [L]/[T], while the right side gives [a][t] = ([L]/[T]^2)([T]) = [L]/[T], confirming consistency. However, dimensional analysis has its limitations. It cannot determine dimensionless constants such as factors of 1/2 or 2pi that may appear in equations. A table showing the seven SI base units in the left column, their symbols in the center column, and common derived units relevant to mechanics in the right column. Below the table, an annotated example of dimensional analysis for the equation x = (1/2)at^2, showing step-by-step how each term's dimensions reduce to [L]. ### III. Significant Figures and Measurement Uncertainty Every measurement has an associated **uncertainty** because no measurement is perfectly exact. Significant figures provide a shorthand for expressing the precision of a measured value. The rules are straightforward: all non-zero digits are significant, zeros between non-zero digits are significant, leading zeros are not significant, and trailing zeros after a decimal point are significant. When performing arithmetic, the rules for significant figures differ depending on the operation. For **multiplication and division**, the result should have the same number of significant figures as the input with the fewest significant figures. For **addition and subtraction**, the result is rounded to the least precise decimal place among the inputs. For error propagation with independent errors, the rules also depend on the operation. For sums and differences, absolute errors add in quadrature: delta_z = sqrt(delta_x^2 + delta_y^2). For products and quotients, relative errors add in quadrature: delta_z/z = sqrt((delta_x/x)^2 + (delta_y/y)^2). Scientific notation is useful for clarifying ambiguity in significant figures, as in the difference between 1500 (ambiguous) and 1.500 x 10^3 (clearly four significant figures). ### IV. Scalars and Vectors A **scalar** is a quantity described by magnitude alone. Examples include mass, temperature, speed, and energy. A **vector**, by contrast, is a quantity described by both magnitude and direction. Displacement, velocity, and force are all vectors. Vectors are typically written with an arrow over the symbol or in boldface (**v**), while the magnitude of a vector is denoted |**v**| or simply v. Graphically, vectors are represented as arrows whose length is proportional to magnitude and whose orientation indicates direction. Panel A: A comparison chart listing common scalars (mass, time, temperature, energy, speed) and common vectors (displacement, velocity, acceleration, force, momentum) with icons for each. Panel B: A graphical depiction of a vector arrow labeled with its magnitude and direction angle relative to the positive x-axis. ### V. Vector Addition and Subtraction Vectors can be added using the **graphical (tip-to-tail) method**, in which the tail of the second vector is placed at the tip of the first, and the resultant vector is drawn from the tail of the first to the tip of the last. Alternatively, the **parallelogram method** places both vectors tail-to-tail, constructs a parallelogram, and takes the diagonal from the common tail as the resultant. Vector subtraction follows naturally: **A** - **B** = **A** + (-**B**), where -**B** has the same magnitude as **B** but points in the opposite direction. Vector addition is commutative (**A** + **B** = **B** + **A**) and associative ((**A** + **B**) + **C** = **A** + (**B** + **C**)). ### VI. Vector Components and Unit Vectors Any vector in two dimensions can be resolved into components along perpendicular axes. If a vector **A** makes angle theta with the positive x-axis, its components are A_x = A cos(theta) and A_y = A sin(theta). Conversely, the magnitude can be recovered from the components as A = sqrt(A_x^2 + A_y^2), and the direction as theta = arctan(A_y / A_x), taking care to identify the correct quadrant. **Unit vectors** are dimensionless vectors of magnitude 1 that point along each coordinate axis: i-hat along x, j-hat along y, and k-hat along z. Any vector can then be expressed as **A** = A_x i-hat + A_y j-hat + A_z k-hat. Adding vectors by components is especially convenient: the resultant **R** = **A** + **B** has components R_x = A_x + B_x and R_y = A_y + B_y. A diagram showing a vector A in the x-y plane making angle theta with the positive x-axis. Dashed lines project A onto the x-axis (labeled A_x = A cos theta) and y-axis (labeled A_y = A sin theta). Unit vectors i-hat and j-hat are shown along the axes. A second smaller diagram shows tip-to-tail addition of two vectors using component method with a right triangle highlighting the resultant. ### VII. Dot Product and Cross Product The **dot product** (or scalar product) of two vectors **A** and **B** is defined as **A** . **B** = AB cos(theta), where theta is the angle between them. In component form, **A** . **B** = A_x B_x + A_y B_y + A_z B_z. The result is a **scalar**. The dot product finds important physical application in the definition of work, W = **F** . **d**. When two vectors are perpendicular, their dot product is zero. The **cross product** (or vector product) is defined as **A** x **B** = AB sin(theta) n-hat, where n-hat is a unit vector perpendicular to the plane containing **A** and **B**, with its direction given by the right-hand rule. The result is a **vector**. In component form, the cross product is computed using the determinant method: (**A** x **B**)_x = A_y B_z - A_z B_y, (**A** x **B**)_y = A_z B_x - A_x B_z, and (**A** x **B**)_z = A_x B_y - A_y B_x. The cross product appears in the definition of torque, tau = **r** x **F**. When two vectors are parallel, their cross product is zero. Unlike the dot product, the cross product is **not commutative**: **A** x **B** = -(**B** x **A**). Panel A: Two vectors A and B with angle theta between them, illustrating the dot product as the projection of A onto B multiplied by B's magnitude. Panel B: Two vectors A and B originating from the same point, with the cross product vector shown perpendicular to the plane formed by A and B. A right hand is depicted showing the right-hand rule: fingers curl from A to B, and the thumb points in the direction of A x B.