Book contents
- Frontmatter
- Contents
- Preface
- 1 Motion on Earth and in the Heavens
- 2 Energy, Heat and Chance
- 3 Electricity and Magnetism
- 4 Light
- 5 Space and Time
- 6 Least Action
- 7 Gravitation and Curved Spacetime
- 8 The Quantum Revolution
- 9 Quantum Theory with Special Relativity
- 10 Order Breaks Symmetry
- 11 Quarks and What Holds Them Together
- 12 Unifying Weak Forces with QED
- 13 Gravitation Plus Quantum Theory – Stars and Black Holes
- 14 Particles, Symmetries and the Universe
- 15 Queries
- APPENDIX A The Inverse-Square Law
- APPENDIX B Vectors and Complex Numbers
- APPENDIX C Brownian Motion
- APPENDIX D Units
- Glossary
- Bibliography
- Index
APPENDIX B - Vectors and Complex Numbers
Published online by Cambridge University Press: 20 January 2010
- Frontmatter
- Contents
- Preface
- 1 Motion on Earth and in the Heavens
- 2 Energy, Heat and Chance
- 3 Electricity and Magnetism
- 4 Light
- 5 Space and Time
- 6 Least Action
- 7 Gravitation and Curved Spacetime
- 8 The Quantum Revolution
- 9 Quantum Theory with Special Relativity
- 10 Order Breaks Symmetry
- 11 Quarks and What Holds Them Together
- 12 Unifying Weak Forces with QED
- 13 Gravitation Plus Quantum Theory – Stars and Black Holes
- 14 Particles, Symmetries and the Universe
- 15 Queries
- APPENDIX A The Inverse-Square Law
- APPENDIX B Vectors and Complex Numbers
- APPENDIX C Brownian Motion
- APPENDIX D Units
- Glossary
- Bibliography
- Index
Summary
This appendix gives a brief explanation of two important mathematical ideas: vectors and complex numbers.
First, vectors. There are many physical quantities that naturally have a direction associated with them, in addition to their magnitude. Examples are velocities, forces, electric fields. These are vectorial quantities, and mathematicians say that they can be “represented by a vector”. But the simplest example of a vector is the geometrical displacement (in a straight line) from one point to another. (The Latin word vector means “carrier”, as in the usage insect vector). We may use this example to illustrate the mathematical properties of vectors.
Mathematicians use symbols to denote vectors, and it is conventional to use boldface type, v, E, and so on, to emphasize that they are not ordinary numbers. Are there mathematical operations that can be performed on vectors, addition, multiplication, and so forth? There is a very simple and natural definition of addition. It is illustrated in Figure A.3.
Any vector (in ordinary three-dimensional space) can be made up as the sum of three vectors, one in each of three specified independent directions. Thus a vector needs three ordinary numbers in order to specify it – the lengths of the three “component” vectors.
Multiplication of vectors is a more complicated question. There is no way to define an operation of “multiplication” acting on a pair of vectors, which has all the properties of multiplication of ordinary numbers (like x × y = y × x).
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- Hidden Unity in Nature's Laws , pp. 437 - 441Publisher: Cambridge University PressPrint publication year: 2001