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A × B
Thumb
Fingers
B
q
A
Figure 1.4
Vector product of A and B .
of the second on an axis which is perpendicular to the first and which lies in the
plane of the two vectors. It is this second geometric interpretation that has most
relevance in dynamics. As we shall see later, moments of force and momentum
involve this type of perpendicular projection. In this topic the vector product will
find its principal application in the study of rotational dynamics.
The scalar and vector products are interesting to us precisely because they have
a geometrical interpretation. That means they represent real things is space. In a
sense, we can think of the scalar product as a machine that takes two vectors as
input and returns a scalar as output. Similarly the vector product also takes two
vectors as its input but instead returns a vector as its output. There are in fact
no other significantly different 4 machines that are able to convert two vectors into
scalar or vector quantities and as a result you will rarely see anything other than
the scalar and vector products in undergraduate/college level physics. There is in
fact a machine that is able to take two vectors as its input and return a new type of
geometrical object that is neither scalar nor vector. We will even meet such a thing
later in this topic when we encounter tensors in our studies of advanced dynamics
and advanced relativity.
1.1.5 Time
We are constantly exposed to natural phenomena that recur: the beat of a pulse;
the setting of the Sun; the chirp of a cricket; the drip of a tap; the longest day of
the year. Periodic phenomena such as these give us a profound sense of time and
we measure time by counting periodic events. On the other hand, many aspects of
the natural world do not appear to be periodic: living things die and decay without
rising phoenix-like from their ashes to repeat their life-cycle; an egg dropped on
the floor breaks and never spontaneously re-forms into its original state; a candle
burns down but never up. There is a sense that disorder follows easily from order,
that unstructured things are easily made from structured things but that the reverse
is much more difficult to achieve. That is not to say that it is impossible to create
order from disorder - you can do that by tidying your room - just that on average
4 i.e. other than trivial changes such as would occur if we choose instead to define the scalar product to
be A · B = λ AB cos θ where λ is a constant. We choose λ = 1 because it is most convenient but any
other choice is allowed provided we take care to revise the geometrical interpretation accordingly.
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