Environmental Engineering Reference
In-Depth Information
(called scalars) we identify vector quantities in this topic by the use of bold font.
When writing vectors by hand it is usual to either underline the vector, or to put
an arrow over the top. Thus
≡
−
a
a
≡
a
.
Use the notation that you find most convenient, but always maintain the distinction
between vector and scalar quantities. In this topic both upper case (
A
) and lower
case (
a
) notion will be used for vectors where
A
is in general a different vector
from
a
. When a vector has zero magnitude it is impossible to define its direction;
we call such a vector the null vector
0
.
1.1.2 Unit vectors
|
|
. To simplify
the notation we shall adopt the convention that vectors are printed in bold and their
magnitudes are indicated by dropping the bold font, thus
a
The length of a vector
a
is known as its magnitude, often denoted
a
≡ |
a
|
. Often we will
separate the magnitude and direction of a vector, writing
a
=
a
a
,
where
a
is the vector of unit magnitude with the same direction as
a
. Unit vectors,
of which
a
is an example, are often used to specify directions such as the directions
of the axes of a co-ordinate system (see below).
1.1.3 Addition and subtraction of vectors
The geometrical rules for adding and subtracting vectors are illustrated in
Figure 1.2. Addition of the vectors
A
and
B
involves sliding the vectors until
they are “head-to-tail”, so that the resultant vector connects the tail of
A
to the
head of
B
. The vector
A
is defined as a vector with the same magnitude, but
opposite direction to
A
. The difference
B
−
−
A
is constructed by adding
B
and
−
A
as shown. Subtraction of a vector from itself gives the null vector:
A
−
A
=
0
.
B
B
A
A
+
B
−
A
B
−
A
Figure 1.2
Adding and subtracting vectors.
