The Timpson Lyceum · University Physics I (2025)

Units and Vectors– Lecture 1 (University Physics I)

Overview

This lecture lays the mathematical groundwork for the entire course. It covers how we measure physical quantities (units, metric prefixes, and the discipline of significant figures) and how we represent quantities that have direction (vectors). By the end you should be able to convert between units by treating conversions as multiplication by 1, report answers with the correct precision, and add, scale, and decompose vectors using components and trigonometry. The vector tools introduced here — components, magnitude, the dot product, and the cross product — are used constantly throughout mechanics.

Key Concepts

The three base units (SI). Everything in this course is built from three quantities and their units:

• Length → meter (m)
• Mass → kilogram (kg)
• Time → second (s)

Metric prefixes scale a unit by powers of ten. The ones used most often:

• nano (n) $= 10^{-9}$
• micro (μ) $= 10^{-6}$ (note: the symbol is the Greek μ, because "m" is already taken by milli)
• milli (m) $= 10^{-3}$
• centi (c) $= 10^{-2}$
• kilo (k) $= 10^{3}$

Dimensional analysis (unit consistency). A conversion factor such as $\frac{5280\ \text{ft}}{1\ \text{mi}}$ equals 1, so multiplying by it changes the units without changing the quantity. Always carry units through a calculation and let them cancel — if the leftover units aren't what you expected, you made a mistake. This is the fastest sanity check available. (Watch cubed units: converting $\text{m}^3$ to $\text{cm}^3$ requires the factor $\left(\tfrac{100\ \text{cm}}{1\ \text{m}}\right)^3$, not just one factor.)

Significant figures communicate how precisely a quantity was measured. Two rules:

• Multiplication / division: the result keeps the fewest significant figures of any factor.
• Addition / subtraction: the result keeps the fewest decimal places of any term.

Scientific notation writes a number as one nonzero digit before the decimal point times a power of ten, e.g. $6235.3 = 6.2353 \times 10^{3}$. The number of digits shown should match the significant figures.

Scalars vs. vectors.

• A scalar is a single real number (e.g. mass, time, temperature).
• A vector is an ordered collection of real numbers — equivalently, a quantity with magnitude and direction. We write it with an arrow, $\vec{A}$, and list components in angle brackets, $\vec{A} = \langle a_x,\ a_y \rangle$ (or $\langle a_x, a_y, a_z\rangle$ in 3D).

Vector operations.

• Addition is component-wise: $\vec{A} + \vec{B} = \langle a_x + b_x,\ a_y + b_y \rangle$. It is commutative, and geometrically it is "tip-to-tail." A vector's position on the page doesn't matter — only its components do.
• Scalar multiplication is component-wise: $c\,\vec{A} = \langle c\,a_x,\ c\,a_y \rangle$.
• Magnitude (length): $\lvert \vec{A} \rvert = \sqrt{a_x^{2} + a_y^{2}}$ (add $a_z^2$ in 3D). This is just the Pythagorean theorem. Note that $\lvert \vec{A} + \vec{B}\rvert \neq \lvert\vec{A}\rvert + \lvert\vec{B}\rvert$ in general — the triangle inequality says the direct path is never longer than the two-leg path.

Components and trigonometry. If $\vec{A}$ has magnitude $A$ and points at angle $\theta$ from the x-axis, then $a_x = A\cos\theta$ and $a_y = A\sin\theta$. Going the other way, the direction is $\theta = \tan^{-1}\!\left(\dfrac{a_y}{a_x}\right)$. Tip: when in doubt, redraw the right triangle so the angle of interest sits at a corner, then read off opposite/adjacent directly.

Unit vectors. $\hat{\imath}, \hat{\jmath}, \hat{k}$ point along $+x, +y, +z$ and have length 1, so any vector can be written $\vec{A} = a_x\hat{\imath} + a_y\hat{\jmath} + a_z\hat{k}$.

Two ways to multiply vectors.

• Dot (scalar) product → produces a scalar: $\vec{A}\cdot\vec{B} = a_x b_x + a_y b_y + a_z b_z$, which also equals $\lvert\vec{A}\rvert\,\lvert\vec{B}\rvert\cos\theta$, where $\theta$ is the angle between the two vectors.
• Cross product → produces a vector perpendicular to both $\vec{A}$ and $\vec{B}$, computed as a determinant: $\vec{A}\times\vec{B} = \begin{vmatrix} \hat{\imath} & \hat{\jmath} & \hat{k} \\ a_x & a_y & a_z \\ b_x & b_y & b_z \end{vmatrix}.$ Its magnitude is $\lvert\vec{A}\times\vec{B}\rvert = \lvert\vec{A}\rvert\,\lvert\vec{B}\rvert\sin\theta$, and its direction is given by the right-hand rule (point your fingers along $\vec{A}$, curl them toward $\vec{B}$, and your thumb points along $\vec{A}\times\vec{B}$).

Worked Example

(a) Unit conversion via dimensional analysis. Convert a land-speed record of $763.0\ \text{mi/h}$ to m/s.

1. Chain conversion factors (each equal to 1) so the unwanted units cancel: $763.0\ \frac{\text{mi}}{\text{h}} \times \frac{5280\ \text{ft}}{1\ \text{mi}} \times \frac{12\ \text{in}}{1\ \text{ft}} \times \frac{2.54\ \text{cm}}{1\ \text{in}} \times \frac{1\ \text{m}}{100\ \text{cm}} \times \frac{1\ \text{h}}{3600\ \text{s}}.$
2. Miles, feet, inches, centimeters, and hours all cancel, leaving m/s.
3. Evaluating gives $341.0\ \text{m/s}$ (4 significant figures, matching $763.0$). A quick sanity check — m/s is roughly half of mi/h — predicted "a few hundred," which fits.

(b) Magnitude and direction of a displacement. A skier travels $1.00\ \text{km}$ north and then $2.00\ \text{km}$ east. How far, and in what direction, is she from the start?

1. The two legs are perpendicular, so the straight-line distance is the hypotenuse: $d = \sqrt{(1.00)^2 + (2.00)^2} = \sqrt{5} \approx 2.24\ \text{km}.$
2. For the direction, measure the angle north of east: $\theta = \tan^{-1}\!\left(\frac{1.00}{2.00}\right) \approx 26.6^{\circ}.$
3. So she is $2.24\ \text{km}$ away, $26.6^{\circ}$ north of east — equivalently $63.4^{\circ}$ east of north (the two angles are complementary, summing to $90^{\circ}$).

Summary

• The course is built on three SI base units — meter, kilogram, second — combined with metric prefixes.
• Treat unit conversions as multiplication by 1 and let units cancel; mismatched leftover units signal an error.
• Keep the fewest significant figures (×, ÷) or the fewest decimal places (+, −) when reporting results.
• A vector carries magnitude and direction; add and scale vectors component-wise, and find length with the Pythagorean theorem.
• Switch between magnitude/angle and components with $a_x = A\cos\theta$, $a_y = A\sin\theta$, and $\theta = \tan^{-1}(a_y/a_x)$.
• The dot product yields a scalar; the cross product yields a vector perpendicular to both inputs.