MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Physics
8.02
Spring 2004
Review A: Vector Analysis
A.1 Vectors………………………………………………………………………….
. A2
A.1.1
Introduction………………………………………………………………. A2
A.1.2
Properties of a Vector…………………………………………………….
. A2
A.1.3
Application of Vectors…………………………………………………….A6
A.2
Dot Product……………………………………………………………….
.. A10
A.2.1
Introduction
...………………………………………………………….
...A
10
A.2.2
Definition……………………………………………………………….
.. A11
A.2.3
Properties of Dot Product……………………………………………….
. A12
A.2.4
Vector Decomposition and the Dot Product ……………………………. A12
A.3
Cross Product …………………………………………………………….
. A14
A.3.1
Definition: Cross Product ………………………………………………. A14
A.3.2
Righthand Rule for the Direction of Cross Product …………………… A15
A.3.3
Properties of the Cross Product ………………………………………… A16
A.3.4
Vector Decomposition and the Cross Product …………………………. A17
A1
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View Full DocumentVector Analysis
A.1 Vectors
A.1.1 Introduction
Certain physical quantities such as mass or the absolute temperature at some point only
have magnitude. These quantities can be represented by numbers alone, with the
appropriate units, and they are called scalars. There are, however, other physical
quantities which have both magnitude and direction; the magnitude can stretch or shrink,
and the direction can reverse. These quantities can be added in such a way that takes into
account both direction and magnitude. Force is an example of a quantity that acts in a
certain direction with some magnitude that we measure in newtons. When two forces act
on an object, the sum of the forces depends on both the direction and magnitude of the
two forces. Position, displacement, velocity, acceleration, force, momentum and torque
are all physical quantities that can be represented mathematically by vectors. We shall
begin by defining precisely what we mean by a vector.
A.1.2 Properties of a Vector
A vector is a quantity that has both direction and magnitude. Let a vector be denoted by
the symbol
. The magnitude of
A
G
A
G
is

A
≡
A 
G
.
We can represent vectors as geometric
objects using arrows. The length of the arrow corresponds to the magnitude of the vector.
The arrow points in the direction of the vector (Figure A.1.1).
Figure A.1.1
Vectors as arrows.
There are two defining operations for vectors:
(1) Vector Addition:
Vectors can be added.
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