1.
Vector Algebra
1.1 Review of Basic Ideas
(p.1 – p.8)
In engineering and science, physical quantities which are completely specified by their magnitude (size)
are known as scalars
. Examples are: mass, temperature, volume, resistance, charge, voltage, current, etc.
Other quantities possess both magnitude and direction are known as vectors
. Examples of vector
quantities are: velocity, acceleration, force, electric field, magnetic field etc and will be denoted by
, etc. Vectors may be represented geometrically by directed line segments. If
A
and
B
are
two geometrical points in
, , , ,
vaFEB
GGJ
GJ
G
3
R
, the directed line segment from
A
to
B
is called the
vector from
A
to
B
and
is denoted
AB
JJJG
.
As the name implies, this is a vector quantity with
direction from
A
to
B
and
magnitude
the distance between
and
B.
The vector
AB
is represented by an arrow from
A
to
B
as shown in the
following figure.
The point
is called the initial point of the vector
AB
,
and
B
is called the terminal point of
AB
.
The
magnitude of the vector
AB
is called its length and is denoted
AB
.
1.
Two vectors
a
and
are
G
b
G
equal
if they have the same magnitude and direction. We write
ab
=
G
G
.
Two vectors can be the same even though the initial points and terminal points are different. For
example
A
BO
P
=
in
2
R
because they have the same length and the same direction (they both
proceed one unit to the left and two units up). Thus the same vector can be translated from one
position to another; what is important is that the length and direction remain the same, and not where
the initial points and terminal points are located. For this reason, we shall often denote vectors as
,
etc. which make no reference to the initial points and terminal points.
, ,
abc
GGG
2.
A vector having the same magnitude as
a
G
but the opposite direction is denoted by
.
a
−
G
1