4 Vector Analysis
A vector carries magnitude and direction together, which makes it the natural language for geometry in three dimensions and for the physical quantities — force, velocity, acceleration — that have both.
The section builds the algebra first. Two products are defined, and the distinction between them is the central idea: the scalar (dot) product returns a number and measures how far two vectors point the same way, while the vector (cross) product returns a vector perpendicular to both and measures how far they fail to be parallel. Between them they answer the questions of angle, projection, area and volume, and they give clean descriptions of lines and planes in space.
The section then puts vectors in motion. A vector function traces a curve as its parameter varies, and differentiating it yields velocity and acceleration. Analysing that motion produces the unit tangent, normal and binormal vectors, the curvature of a space curve, and the resolution of acceleration into tangential and normal components — the last of which explains why a body moving at constant speed around a bend is nevertheless accelerating.
4.2 Vectors
4.3 Direction of a Vector
4.4 The Scalar or Dot Product of Two Vectors
4.5 Projections
4.6 Lines in Three Dimensional Space
4.7 The Vector Product of Two Vectors
4.8 Triple Products
4.9 Planes
4.10 Vector Functions and their Derivatives
4.11 Derivatives and Integrals of Vector Functions
4.12 Arc Length
4.13 Curvature
4.14 The Normal and Binormal Vectors
4.15 Motion in Space: Velocity and Acceleration
4.16 Tangential and Normal Components of Acceleration
4.17 Practice Problems
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