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A scalar quantity is invariant under all possible rotational transformations.
The individual components of a vector are not scalars because they change under
transformation. Can we form a scalar out of some combination of the components
of one, or more, vectors? Suppose that we were to define the
``ampersand'' product,

(16) 
for general vectors and . Is
invariant under transformation, as must be the case if it is a scalar number?
Let us consider an example. Suppose that
and
. It is easily seen that
. Let
us now rotate the basis through about the axis. In the new
basis,
and
, giving
. Clearly,
is not invariant under rotational transformation, so
the above definition is a bad one.
Consider, now,
the dot product or scalar product:

(17) 
Let us rotate the basis though degrees about the axis. According to
Eqs. (10)(12), in the new basis
takes the form
Thus,
is invariant under rotation about the axis. It can easily
be shown that it is also invariant under rotation about the  and axes.
Clearly,
is a true scalar, so the above definition is
a good one. Incidentally,
is the only
simple combination of
the components of two vectors which transforms like a scalar. It is easily
shown that the dot product is commutative and distributive:
The associative property is meaningless for the dot product, because we cannot
have
, since
is scalar.
We have shown that the dot product
is coordinate independent.
But what is the physical significance of this? Consider the special case
where
. Clearly,

(20) 
if is the position vector of relative to the origin .
So, the invariance of
is equivalent to the invariance
of the length, or magnitude, of vector under transformation. The length of
vector is usually denoted (``the modulus of '') or sometimes
just , so

(21) 
Figure 5:

Let us now investigate the general case. The length squared of (see Fig. 5) is

(22) 
However, according to the ``cosine rule'' of trigonometry,

(23) 
where denotes the length of side . It follows that

(24) 
Clearly, the invariance of
under transformation is equivalent
to the invariance of the angle subtended between the two vectors. Note that
if
then either , , or the vectors
and are perpendicular. The angle subtended between two vectors
can easily be obtained from the dot product:

(25) 
The work performed by a constant force moving an object through a displacement
is the product of the magnitude of times the displacement in the direction
of . If the angle subtended between and is then

(26) 
The rate of flow of liquid of constant velocity through a loop of vector area
is the product of the magnitude of the area times the component of the
velocity perpendicular to the loop. Thus,

(27) 
Next: The vector product
Up: Vectors
Previous: Vector areas
Richard Fitzpatrick
20060202