Chapter 4 · Introduction

Chapter 4 — Vectors in ℝⁿ

Forces, velocities, GPS positions and rows of data are all vectors. Add them, scale them, measure their lengths and angles, and use them to describe lines and planes in space.

A delivery drone leaves a warehouse in Dubai. Its motors push it through the air at $100$ km/h, but the air itself is moving — a $30$ km/h wind blowing in from the Gulf. Where does the drone actually go, how fast, and which way should it point so that it arrives where it should? The answer is a sum of two arrows. The same arithmetic tells an engineer whether three cables holding a bridge deck are in balance, tells your phone's GPS how far you are from the next turn, and tells a hospital's software which past patient's record is most like today's. Forces, velocities, positions and data records all have several numbers that belong together: they are vectors.

In Chapters 1–3 vectors appeared quietly as the columns of matrices. This chapter puts them in the spotlight. We start with arrows in the plane and in space, translate them into lists of numbers that work in any number of dimensions, and then learn two products. The dot product turns geometry into arithmetic — length, distance, angle and perpendicularity all come from one formula — and it powers applications from the work done by a force to the cosine similarity used by search engines. The cross product, special to three dimensions, builds a vector perpendicular to two others, which is exactly what we need to describe planes, areas and volumes. Everything here is preparation for Chapter 5, where the idea of "things you can add and scale" becomes the abstract notion of a vector space.

Arrows and lists of numbers→Add, scale, combine (4.1)→Dot product: length, angle, projection (4.2)→Cross product: area, normal (4.3)→Lines and planes in space (4.3)
🎯 By the end of this chapter you can…
  • move between arrows and components, compute $\mathbf{u}+\mathbf{v}$, $c\mathbf{v}$ and $\mathbf{u}-\mathbf{v}$ in $\mathbb{R}^n$, and explain them with the triangle and parallelogram laws;
  • decide whether a vector is a linear combination of given vectors, and read a linear system as a vector equation;
  • compute dot products, norms, unit vectors, distances and angles, and test vectors for orthogonality;
  • use the Cauchy–Schwarz and triangle inequalities, split a vector into projection plus orthogonal part, and apply the dot product to work and cosine similarity;
  • compute cross products and use them for normals, areas of parallelograms and triangles, and (with the triple product) volumes;
  • write equations of lines and planes, find their intersections, and compute distances from a point to a line or a plane.
Teaser — Flying a Drone through the Wind

The drone flies at $100$ km/h through the air in the direction of its heading; the wind carries the air (and the drone) with it. Its velocity over the ground is the vector sum air velocity + wind = ground velocity, drawn tip to tail at the start. The target is $200$ km due north. Play launches the drone; turn the heading into the wind until it lands on the target. (Headings are measured from north, positive to the east.)

30
0°
ground velocity (km/h)—
ground speed $\|\mathbf{g}\|$—
time to reach the target line—
miss distance—

Sections in this chapter

4.1

Vectors in ℝ², ℝ³ and ℝⁿ

Arrows and components, addition and scalar multiples, linear combinations, standard unit vectors, and a linear system as one vector equation.

$x_1\mathbf{a}_1+\dots+x_n\mathbf{a}_n=\mathbf{b}$
Open section →4.2

Dot Product, Length and Angle

Norms, unit vectors, distance, angles, orthogonality, Cauchy–Schwarz, projections, work and cosine similarity.

$\mathbf{u}\cdot\mathbf{v}=\|\mathbf{u}\|\|\mathbf{v}\|\cos\theta$
Open section →4.3

Cross Product, Lines and Planes

A vector perpendicular to two others; areas, volumes, lines and planes in space, intersections and distances.

$\mathbf{n}\cdot(\mathbf{x}-\mathbf{p})=0$
Open section →
✅ Before you start
You need coordinates in the plane and Pythagoras' theorem from school, and the basic trigonometry of a right triangle ($\sin$, $\cos$ and $\cos^{-1}$ on a calculator). From this course you need to solve small linear systems by row reduction (1.1 and 1.2), because deciding whether a vector is a linear combination of others is exactly that. For the cross product and the triple product in Section 4.3 you need $2\times2$ and $3\times3$ determinants by cofactor expansion (3.1); the link between determinants, area and volume (3.3) is helpful but not required.
← 3.3 Cramer's Rule, the Adjoint, Area and Volume4.1 Vectors in ℝ², ℝ³ and ℝⁿ →