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Mathematics · Ch 6 — Applications of Vector Algebra

Introduction

6.1

Introduction

Vectors first appeared in your Class XI work as directed quantities — objects with both a magnitude and a direction, written a⃗\vec a or, in component form, a1i^+a2j^+a3k^a_1\hat i+a_2\hat j+a_3\hat k. The word itself comes from the Latin vectus, "to carry." Two vectors with the same magnitude and direction are always equal, regardless of where their initial points sit. The idea grew into today's rigorous theory through Caspar Wessel (1745–1818) and Jean-Robert Argand (1768–1822), who represented a complex number as a directed line segment in the plane — the same picture later generalised by Hermann Grassmann, William Rowan Hamilton, William Kingdon Clifford and Josiah Willard Gibbs into the modern algebra of dot and cross products.

This chapter has two goals. First, to revisit vectors geometrically — as directed line segments AB⃗\vec{AB} — with a little more care than before, because that geometric picture is exactly what is needed to write down the equations of straight lines and planes in three-dimensional space, R3\mathbb{R}^3. Second, to see vector algebra actually at work:

  • the scalar triple product computes the volume of a parallelepiped;
  • the dot product and cross product compute work done and torque in mechanics;
  • vector algebra combined with calculus gives curl and divergence, used in electromagnetism, fluid flow (hydrodynamics, blood flow) and even rocket/satellite trajectories;
  • the dot and cross product together find the distance and angle between the flight paths of two aircraft;
  • a simple dot-product calculation decides how to tilt a solar panel for maximum power, given the sun's direction, and how much power it generates;
  • vector algebra measures angles and distances in satellite-panel layouts, pipe networks, and civil-engineering beam structures.
Note

Every one of these applications ultimately reduces to just three operations you already know — vector addition, the dot product, and the cross product — used with a bit more geometric rigor than before.

Learning Objectives

By the end of this chapter you should be able to:

  • apply the scalar (dot) and vector (cross) products of two and three vectors;
  • solve problems in geometry, trigonometry, and physics using vectors;
  • derive the equation of a line in parametric, non-parametric (vector), and Cartesian form, in different situations;
  • derive the equation of a plane in parametric, non-parametric (vector), and Cartesian form, in different situations;
  • find the angle between two lines and the distance between skew lines; and
  • find the coordinates of the image of a point.
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