Mathematics · Ch 8 — Vector Algebra-I
Introduction
Introduction
Why Vectors?
A pilot planning a flight has to head the aircraft into the wind at just the right angle so that the wind's own push is exactly counteracted and the plane still reaches its destination — done either with a navigation computer or, in its absence, by hand with a working knowledge of vectors. In symbols: if is the aircraft's own velocity and is the wind's velocity, the plane's actual velocity over the ground is — and a natural question follows: in which direction should the aircraft head in order to fly due west?
A skydiver in mid-fall feels two forces at once: gravity pulling straight down, and air resistance pushing up and off to some other angle. The net force is — not obvious until the two are added as vectors, not as plain numbers.
A GPS receiver, too, works by combining vectors to fix a position on the earth, in the air, or on water.
This chapter builds the algebra needed to answer questions like these: how do you add two vectors that point in different directions? How do you describe a vector's direction precisely, using numbers? And what does it mean to 'multiply' two vectors — is the answer a number or another vector? By the end of the chapter you will be able to resolve any vector into components along coordinate axes, add and subtract vectors geometrically and algebraically, and compute two genuinely different kinds of vector product: the scalar (dot) product and the vector (cross) product.
The mathematics of vectors, as we use it today, grew out of the work of the German mathematician H. G. Grassmann (1809–1877) — a secondary-school teacher — and the Irish mathematician W. R. Hamilton (1805–1865), who held high academic office. Hamilton coined the word vector itself, from the Latin for 'to carry.' Later, the American mathematician J. W. Gibbs and the English scientist Oliver Heaviside united the best features of quaternion calculus and Cartesian geometry into the new subject of vector algebra — the form of it we use today first took shape in Gibbs's own teaching notes for his students at Yale. It was a third mathematician, W. K. Clifford (1845–1879), who — in his Elements of Dynamics (1878) — split the old 'quaternion product' into the two distinct vector products this chapter covers: the scalar product and the vector product.