Vectors — the language of motion in 2D and 3D

📚 Reference: Halliday, Resnick, Krane — Physics (4th ed.), Vol 1, Chapter 3.
Independent treatment; no text or figures reproduced. All scenarios are original.

What this chapter is about

In Chapter 2 we studied only 1D motion — a sign + or - was enough. But nature is three-dimensional — a projectile in flight, a satellite in orbit, an electron in a magnetic field. To describe such motion we need quantities that carry both magnitude and direction. Those are vectors.

Chapter 3 builds the language used throughout physics. Not a peripheral tool — vectors are the core language of mechanics, electromagnetism, quantum mechanics.

Sections

1. 3.1 — Scalars and vectors2. 3.2 — Vector addition — graphical method3. 3.3 — Components and unit vectors4. 3.4 — Adding vectors by components5. 3.5 — Scalar (dot) product6. 3.6 — Vector (cross) product7. 3.7 — Worked problems8. 3.8 — Practice problems9. 3.9 — Further reading10. 3.10 — Q&A

Prerequisites

• Basic trigonometry — sin, cos, tan, Pythagorean theorem. If shaky, do a quick review
• Coordinate geometry — Cartesian system (x, y, z)
• 1D kinematics (Chapter 2) — we’re generalizing to 2D/3D

No calculus required — vectors work with algebra. Calculus enters in Chapter 4 onward.

Reading suggestion

• §3.1-3.4 are the foundation for all later physics — read these carefully
• §3.5-3.6 (dot and cross products) look heavy at first but are unavoidable in electromagnetism (§29 Maxwell), rotational mechanics (§10-11), and quantum
• If you only have a few hours: master §3.3 and §3.4 — every 2D/3D problem builds on them

Why vectors matter — a simple example

Two people pull a barrel with ropes:
– Person 1 with 50 N to the north
– Person 2 with 50 N to the east

What’s the total force? Answer: 50 + 50 = 100 N? No! The total is about 70.7 N (pointing northeast). You can’t add these as scalars — the directions differ. Vector language handles this exactly.

Connection to later chapters

• Chapter 4: 2D/3D motion — vector kinematics, projectiles, circular motion
• Chapter 5-6: dynamics — Newton’s laws are all vector equations (\vec F = m \vec a)
• Chapter 10-11: rotation — torque is a cross product
• Chapter 29-32: electromagnetism — electric and magnetic fields are vector; Maxwell’s equations are vector dot/cross products

📖 Open reference: OpenStax University Physics Vol 1 — Chapter 2: Vectors.
📖 Feynman Lectures Vol I — Ch 11: Vectors.