The Chemistry Maths Book, Second Edition

16.11 Exercises 471

6

A detailed treatment of n-dimensional vector spaces was given by Hermann Günther Grassmann (1809–1877),

German mathematician, in his Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik(The theory of linear

extension, a new branch of mathematics) of 1862. The work included the algebras of Hamilton’s quaternions and

Gibbs’ vectors as special cases, but was largely unknown during his lifetime.

16.10 Vector spaces

The vectors discussed in this chapter are three-dimensional vectors, or vectors in

a three-dimensional vector space. The concept of vector can be extended to any

number of dimensions by defining vectors in ndimensions as quantities that have

ncomponents and that obey the laws of vector algebra described in Sections 16.2

and 16.3. In particular, an n–dimensional vector space can be defined by means of n

orthogonal unit vectors,

e

1

1 = 1 (1, 0, 0, =, 0) e

2

1 = 1 (0, 1, 0, =, 0) - e

n

1 = 1 (0, 0, 0, =, 1) (16.80)

Every vector in the space can then be expressed as a linear combination of these,

a 1 = 1 (a

1

, a

2

, a

3

, =, a

n

) 1 = 1 a

1

e

1

1 + 1 a

2

e

2

1 + 1 a

3

e

3

1 +-+ 1 a

n

e

n

(16.81)

These quantities obey the rules of vector algebra.

6

For the vector space so defined, an inner (scalar) productis associated with each

pair of vectors,

a 1

·

1 b 1 = 1 a

1

b

1

1 + 1 a

2

b

2

1 + 1 a

3

b

3

1 +-+ 1 a

n

b

n

1 = 1 b 1

·

1 a (16.82)

The inner product is often denoted by (a, 1 b). Two vectors are orthogonal

(‘perpendicular’) if their inner product is zero, and the length or normof a vector is

(16.83)

A vector space with these properties is called an inner product space. The particular

space described in this section is the n–dimensional Euclidean spaceR

n

16.11 Exercises

Section 16.2

1.The position vectors of the points A, B, C and D are a, b, c, and d. Express the following

quantities in terms of a, b, c, and d: (i) and , (ii)the position of the centroid of

the points, (iii)the position of the midpoint of , (iv)the position of an arbitrary point

on the line (the equation of the line).

Section 16.3

2.Two sides of the triangle ABC are and. Find

Find (i)the vectora 1 = 1 (a

1

, a

2

, a

3

)with the given initial pointP(x

1

, y

1

, z

1

)and terminal point

Q(x

2

, y

2

, z

2

), (ii)the length of a, (iii)the unit vector parallel to a.

3.P(1, −2, 0), Q(4, 2, 0) 4.P(−3, 2, 1), Q(−1, −3, 2) 5.P(0, 0, 0), Q(2, 3, 1)

BC

AC

AB =,,()320

=,,−()21 1

BC

BC

BA

AB

||a aaa a

n

=aa 11 = ++++

·

1

2

2

2

3

22