Chapter 0: Vectors in ℝⁿ: Arrows, Lengths, and Angles

Everything in this book is built from one kind of object: a list of numbers, read as a point or as an arrow. A pair $(x, y)$ names a point of the plane; a triple $(x, y, z)$ names a point of space; and a list of $n$ numbers names a point of $\mathbb{R}^n$, a space we cannot draw but can compute in exactly as easily. The same list, read as a displacement — go this far in the first direction, this far in the second — is a vector, and vectors are the currency of linear algebra: they are what we add, scale, project, and, later, feed to matrices.

This chapter lays out that currency before any equation is solved. It begins with points and the distance between them, which turns out to be a Pythagorean fact applied twice. It then introduces vectors as arrows with components, and the two operations every later chapter assumes: adding two arrows head to tail, and stretching one by a number. From those come unit vectors — arrows of length exactly $1$ that carry a direction and nothing else — and two products that measure how a pair of vectors sits in space: the dot product, which reads off the angle between them and detects right angles, and the cross product, which stands perpendicular to both and measures the area they span. The chapter closes with the two flat objects of $\mathbb{R}^3$, lines and planes, each described by a point and a vector.

Every section carries a figure you can drive. You will drop a pair of points to the floor and watch the distance formula assemble itself from two right triangles; drag the corners of a parallelogram and see the sum of two vectors track its diagonal; swing a vector around a sphere while its unit vector rides the surface; open and close the angle between two arrows until their dot product changes sign; tilt a parallelogram out of the floor and watch its cross product lean to stay perpendicular; and slide a plane along its own normal. The formulas are short. The point of the chapter is that each of them is a picture you have already seen move.