3.5 — The scalar (dot) product
There are two ways to multiply vectors — both physically useful. The scalar (dot) product returns a number — a scalar. Used whenever “amount of…
3.4 — Adding vectors by components
With the components tool from §3.3, vector addition reduces to simple scalar addition per axis. More accurate than graphical and scalable to many vectors.
3.3 — Components and unit vectors
The graphical method (§3.2) is fine but doesn’t scale. Components are the algebraic tool that makes vector work scalable — 100 vectors as easy as 2.…
3.2 — Vector addition — the graphical method
Adding vectors is not like adding numbers — magnitudes AND directions must combine. The graphical method is direct, intuitive, and the best way to…
3.1 — Scalars and vectors
Physics deals with two kinds of quantities — scalars and vectors. The distinction is simple, but its consequences are deep: from notation, to how you…
Chapter 3 — Vectors — the language of motion in 2D and 3D
Vectors are the language of 2D and 3D motion — vector addition, components, unit vectors, dot product, and cross product with full practice sets.
How Galileo measured g — inclined planes, water clocks, and pendulums
How Galileo in 1600 measured gravitational acceleration g using inclined planes, water clocks, and pendulums — with no digital stopwatch. Includes an SVG animation of the inclined-plane experiment.
The definite integral for physicists — from Riemann sums to real applications
The definite integral for physicists — from Riemann sums to practical applications: displacement, work, potential energy. Step-by-step guide for students new to calculus.
2.10 — Q&A — Frequently asked questions for Chapter 2
Common questions students raise while working through Chapter 2. Each answer is short and direct; links point to the relevant section for deeper reading.
2.9 — Further reading and references
Chapter 2 built a foundational toolkit. To go deeper or see different perspectives, these are good resources.