Environmental Engineering Reference
In-Depth Information
1.1.4 Multiplication of vectors
There are two types of vector multiplication that are useful in classical physics.
The scalar (or dot) product of two vectors
A
and
B
is defined to be
A
·
B
=
AB
cos
θ,
(1.1)
This scalar quantity (a pure number) has a simple geometrical interpretation.
It is the projection of
B
on
A
,i.e.
B
cos
θ
, multiplied by the length of
A
(see
Figure 1.3). Equally, it may be thought of as the projection of
A
on
B
,i.e.
A
cos
θ
,
multiplied by the length of
B
. Clearly the scalar product is insensitive to the order
of the vectors and hence
A
A
. The scalar product takes its maximum
value of
AB
when the two vectors are parallel, and it is zero when the vectors
are mutually perpendicular.
·
B
=
B
·
A
cos q
B
q
B
cos q
A
Figure 1.3 Geometry of the scalar product.
A
·
B
is the product of the length of
A
,andthe
projection of
B
onto
A
or alternatively the product of the length of
B
, and the projection of
A
onto
B
.
The vector (or cross) product is another method of multiplying vectors that is
frequently used in physics. The cross product of vectors
A
and
B
is defined to be
A
×
B
=
AB
sin
θ
n
,
(1.2)
where
θ
is the angle between
A
and
B
and
n
is a unit vector normal to the
plane containing both
A
and
B
.Whether
n
is 'up' or 'down' is determined by
convention and in our case we choose to use the right-hand screw rule; turning
the fingers of the right hand from
A
to
B
causes the thumb to point in the sense
of
n
as is shown in Figure 1.4. Interchanging the order of the vectors in the
product means that the fingers of the right hand curl in the opposite sense and the
direction of the thumb is reversed. So we have
B
×
A
=−
A
×
B
.
(1.3)
The magnitude of the vector product
AB
sin
θ
also has a simple geometrical
interpretation. It is the area of the parallelogram formed by the vectors
A
and
B
.
Alternatively it can be viewed as the magnitude of one vector times the projection
