The Chemistry Maths Book, Second Edition

456 Chapter 16Vectors

(iv) The velocity in the x-direction isv

x

1 = 14 so that the body is moving in the positive

x-direction with constant speed 4. (v) The acceleration is in the y-direction with

a

y

1 = 1 d

2

y 2 dt

2

1 = 1 − 4 yand, by equation (12.29) in Section 12.5, the motion is simple

harmonic in this direction. (vi) The body moves to the right with constant speed

v

x

1 = 14 whilst undergoing simple harmonic motion in the perpendicular y-direction,

as shown in Figure 16.18.

0 Exercises 17, 18

16.5 The scalar (dot) product

The application of vectors to physical problems has led to the definition of two ways

of multiplying a vector by a vector; one way gives a scalar as product, the other gives a

vector product (Section 16.6).

4

The scalar productor dot productof aand bis defined as

a 1

·

1 b 1 = 1 ab 1 cos 1 θ (16.30)

(read as ‘a dot b’) wherea 1 = 1 |a|andb 1 = 1 |b|are the

lengths of the vectors and θ(0 1 ≤ 1 θ 1 ≤ 1 π)is the angle

between the directions of aandb. In terms of

components, ifa 1 = 1 (a

x

, a

y

, a

z

)andb 1 = 1 (b

x

, b

y

, b

z

)then

a 1

·

1 b 1 = 1 a

x

b

x

1 + 1 a

y

b

y

1 + 1 a

z

b

z

(16.31)

Figure 16.18

4

Clerk Maxwell used Hamilton’s quaternions in the Treatise on Electricity and Magnetismof 1873, but it was

observed, independently, by Heaviside and Gibbs that the full algebra of quaternions was not necessary for the

representation of physical quantities.

Oliver Heaviside (1850–1925), English electrical engineer, is noted for his development of Laplace transform

techniques. Josiah Willard Gibbs (1839–1903), American engineer, chemist, and ‘among the most renowned of

theoretical physicists of all time’ (Max Planck), was born, lived, and died in New Haven, Connecticut. Gibbs is best

known for his work in thermodynamics and statistical mechanics. His introduction of the concept of chemical

potential in On the equilibrium of heterogeneous substancesin 1876–8 marks the foundation of modern chemical

thermodynamics. He developed the modern notation and the concepts of scalar and vector products in the early

1880’s to simplify the mathematical treatment in Maxwell’s Treatise. The unpublished work, circulated as lecture

notes, was popularized by Edwin B. Wilson’s Vector analysis, founded upon the lectures of J. Willard Gibbsin 1901.

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