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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
``percent'' product,

(1283) 
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 coordinate axes through about . In the new
coordinate system,
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:

(1284) 
Let us rotate the coordinate axes though degrees about . According to
Equations (A.1280)(A.1282),
takes the form
in the new coordinate system.
Thus,
is invariant under rotation about . It can easily
be shown that it is also invariant under rotation about and .
We conclude that
is a true scalar, and that 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: i.e.,
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 geometric significance of this? Well, in the special case
where
, we get

(1287) 
So, the invariance of
is equivalent to the invariance
of the magnitude of vector under transformation.
Let us now investigate the general case. The length squared of in the
vector triangle shown in Figure A.101 is

(1288) 
However, according to the ``cosine rule'' of trigonometry,

(1289) 
where denotes the length of side . It follows that

(1290) 
In this case, 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 mutually perpendicular. The angle subtended between two vectors
can easily be obtained from the dot product: i.e.,

(1291) 
Figure A.101:
A vector triangle.

The work performed by a constant force which moves 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

(1292) 
The work performed by a nonconstant force which moves an
object through an infinitesimal displacement in a time interval is
. Thus, the rate at which the force does
work on the object, which is usually referred to as the power, is
, or
,
where
is the object's instantaneous velocity.
Next: Vector Product
Up: Vector Algebra and Vector
Previous: Coordinate Transformations
Richard Fitzpatrick
20110331