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\begin{document}
ANY \htmladdnormallink{magnitude}{http://planetphysics.us/encyclopedia/AbsoluteMagnitude.html} which has \emph{size}, in the ordinary \htmladdnormallink{algebraic}{http://planetphysics.us/encyclopedia/CoIntersections.html} sense of
the word, as well as \emph{direction} in space, is termed a \htmladdnormallink{vector}{http://planetphysics.us/encyclopedia/Vectors.html}, whereas
the common algebraic magnitudes, which have nothing to do with
direction in space, which have no directional properties, but are
each determined completely by a single (real) number, are called
\htmladdnormallink{scalars}{http://planetphysics.us/encyclopedia/Vectors.html}. The typical case of a vector and, in fact, the intuitional
representative of any vector, is a segment of a straight line of some
definite length and of some definite direction in space, the size
of the vector being represented by the length, and its direction by
the direction of the straight line.
Thus, the displacement of a \htmladdnormallink{particle}{http://planetphysics.us/encyclopedia/Particle.html} from some initial to some
other final \htmladdnormallink{position}{http://planetphysics.us/encyclopedia/Position.html} is a vector, and is represented by the segment
of the straight line joining the two positions and directed from the
first to the second. Other examples of vectors are the instantaneous
\htmladdnormallink{velocity}{http://planetphysics.us/encyclopedia/Velocity.html} of a particle, its \htmladdnormallink{acceleration}{http://planetphysics.us/encyclopedia/Acceleration.html}, its \htmladdnormallink{momentum}{http://planetphysics.us/encyclopedia/Momentum.html}, the force
acting on a particle, also the instantaneous rotational velocity of,
say, a \htmladdnormallink{rigid body}{http://planetphysics.us/encyclopedia/CenterOfGravity.html} round a given axis, and so on. On the other
hand, \htmladdnormallink{mass}{http://planetphysics.us/encyclopedia/CosmologicalConstant2.html} (in \htmladdnormallink{classical mechanics}{http://planetphysics.us/encyclopedia/MathematicalFoundationsOfQuantumTheories.html}), \htmladdnormallink{temperature}{http://planetphysics.us/encyclopedia/BoltzmannConstant.html}, vis viva, \htmladdnormallink{energy}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} in general; gravitational, electric or magnetic potential (of fixed \htmladdnormallink{charges}{http://planetphysics.us/encyclopedia/Charge.html} or magnets), mechanical or any other kind of \htmladdnormallink{work}{http://planetphysics.us/encyclopedia/Work.html} are
all scalars.
The size of a vector, or magnitude (absolute value) apart from
direction, is called its \htmladdnormallink{tensor}{http://planetphysics.us/encyclopedia/Tensor.html}, or sometimes intensity. Thus, the
tensor of a vector is an essentially \emph{positive scalar}.
Every vector can be determined completely by \emph{three} scalar
quantities, for instance, by its projections on any three fixed axes,
orthogonal or oblique, but not coplanar, these projections being
commonly called the vector's components; for example, the
components of a force or the components of a velocity. We also
may use polar coordinates, that is to say, we may define the tensor
of a vector by the scalar $r$, and the direction by two other scalars,
\emph{i.e.} by two angles $\theta$, $\phi$, say the geographical latitude and longitude.
In this way we get again three mutually independent scalars
determining a single vector.
Obviously, such a decomposition of a vector into its three components
or, more generally, into three mutually independent scalars,
will in the majority of cases bring in some artificial elements,
especially if the \htmladdnormallink{system}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html} of reference (axes, etc.) or the scaffolding
constructed round the natural entities or phenomena be chosen
quite at random without having anything in common with the
essential characters of these entities or phenomena. Very often
such a procedure gives rise to a hopeless complication of the
resulting scalar formulae, a complication which does not arise
from the intrinsic peculiarities of the phenomena in question,
but is wholly artificial, a complication not due to Nature but
to the (mathematizing) naturalist. Now, Nature is of herself wonderfully
complicated; so that supplementary complication is not
wanted.
This remark alone may suggest that to operate with vectors, each
taken as a whole, without decomposing them into scalar components,
may be more convenient and more simple, especially in those regions
of research in which we are concerned \emph{mainly with vectors} or directed
magnitudes, as in \htmladdnormallink{Electromagnetism}{http://planetphysics.us/encyclopedia/Electromagnetism.html} and in General \htmladdnormallink{mechanics}{http://planetphysics.us/encyclopedia/Mechanics.html}.
But a true appreciation of the advantage of the vector method over
the Cartesian (or scalar component) procedure is possible only when
we see it actually at work, and the main \htmladdnormallink{object}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} of each of the
following chapters is to exhibit this working in Mechanics. Still
more conspicuous is the service done by the vector method in
Electromagnetism, especially in the hands of Oliver Heaviside, to
whom also is due that simplified form of this mathematical method,
which in its main features we shall now develop.
{\bf Definition I.} By saying that two vectors are \emph{equal to one another}
we mean that their \emph{tensors} are equal and that they have the same
\emph{direction}, or, what is the same thing, that their representative
straight line-segments have the same lengths and are parallel to
one another and similarly (not oppositely) directed; but the
equality is independent of their position in space.
According to this definition, the shifting of a given vector parallel
to itself is quite immaterial, or does not change the vector.
Thus, all the vectors represented on Fig. I. are to be considered
as equal to one another. The parallel shifting of a vector, which
by convention leaves it the same, is, of course, not confined to one
plane.
\begin{figure}
\includegraphics[scale=.8]{FigI.eps}
\end{figure}
Following the example of Heaviside and Gibbs vectors will be
printed in {\bf bold}, and their tensors will be denoted
by the same letters printed in ordinary \htmladdnormallink{type}{http://planetphysics.us/encyclopedia/Bijective.html} (or simple \emph{italics}).
Thus
$$A, B, C$$
will be the tensors of the vectors
{\bf A}, {\bf B}, {\bf C}
respectively.
If the tensor of a vector, say ${\bf a}$, be equal to \emph{unity} (in a given
scale), i.e. if
$$a=1$$
then the vector ${\bf a}$ is called a {\bf unit-vector}.
By the definition, every tensor is an absolute or \emph{positive} number.
It has, of course, the same denomination as the physical, or geometrical,
quantity represented by the vector, i.e. if {\bf A} be a velocity,
then $A$ signifies so many centimetres per second, and similarly in
all other cases.
We pass now to the fundamental \htmladdnormallink{operations}{http://planetphysics.us/encyclopedia/Cod.html} of vector algebra.
These are : the \emph{addition} of two vectors, and its inverse, the
\emph{subtraction} of one vector from another, and two different kinds
of \emph{multiplication}, the scalar and the vector multiplication of two
vectors (\htmladdnormallink{scalar product}{http://planetphysics.us/encyclopedia/DotProduct.html} and \htmladdnormallink{vector product}{http://planetphysics.us/encyclopedia/VectorProduct.html}). (The division, i.e. the quotient
of two vectors, belongs to the Calculus of Quaternions, due to Hamilton,
and has nothing to do with Heaviside's and Gibbs' vector method to be developed
here, notwithstanding that the latter has grown out of the former,
historically.)
Let us begin with the operation of addition and its result, the
\emph{sum} of two vectors.
{\bf Definition II.} If the end of the vector ${\bf A}$ coincides with the
beginning of another vector ${\bf B}$, then we call {\bf sum} of ${\bf A}$ and ${\bf B}$
and denote by
$${\bf A} + {\bf B}$$
a third vector ${\bf R}$ which runs from the beginning of ${\bf A}$ to the end
of ${\bf B}$ (Fig. 2).
\begin{figure}
\includegraphics[scale=.8]{Fig2.eps}
\end{figure}
This definition of sum seems at first too narrow, as far as it
appeals to the chain-arrangement of the two vectors; but in fact it
embraces the \htmladdnormallink{concept}{http://planetphysics.us/encyclopedia/PreciseIdea.html} of the sum of \emph{any} two vectors. For, if ${\bf B}$ or
its representative line be originally given in a quite arbitrary manner
relatively to ${\bf A}$, we can always shift it parallel to itself (which is
allowed by Definition I.) till its beginning is brought into coincidence
with the end of ${\bf A}$.
For the same reason we see that the sum of two vectors ${\bf A}$, ${\bf B}$
starting from the same origin $O$ is given by the \emph{diagonal} OP of
the \emph{parallelogram} constructed on the addends ${\bf A}$, ${\bf B}$ (Fig. 3). For,
by Def. I., ${\bf B'} = {\bf B}$, since $B' = B$ and ${\bf B} || {\bf B}$ (i.e. ${\bf B'}$
parallel to and concurrent with ${\bf B}$), in Euclidean space, of course.
\begin{figure}
\includegraphics[scale=.8]{Fig3.eps}
\end{figure}
Again, in the same parallelogram, ${\bf A'} = {\bf A}$ (since $A' = A$ and ${\bf A} || {\bf A}$),
and therefore
$${\bf B} + {\bf A} = {\bf B} + {\bf A'} = {\bf R} = {\bf A} + {\bf B'} = {\bf A} + {\bf B} $$
hence, for any two vectors,
$${\bf A} + {\bf B} = {\bf B} + {\bf A}$$
Now, the sum of two vectors being again a vector, ${\bf A} + {\bf B} = {\bf R}$,
we can add to ${\bf R}$ any third vector, thus getting
$${\bf R} + {\bf C} = \left ( {\bf A} + {\bf B} \right ) + {\bf C} = {\bf C} + \left ({\bf A} + {\bf B} \right) $$
Again, arranging ${\bf A}$, ${\bf B}$, ${\bf C}$ in a \emph{chain}, i.e. so
that the end of ${\bf A}$ is the beginning of ${\bf B}$, the end of ${\bf B}$
the beginning of ${\bf C}$, we see at once (Fig. 4) that
$${\bf A} + {\bf B} + {\bf C} = \left ( {\bf A} + {\bf B} \right ) + {\bf C} = {\bf A} + \left ( {\bf B} + {\bf C} \right ) $$
\begin{figure}
\includegraphics[scale=.8]{Fig4.eps}
\end{figure}
the result being always the same, namely to get from the beginning
of ${\bf A}$ to the end of ${\bf C}$. The same thing is true for the sum of four,
five and more vectors. Thus we get the following \htmladdnormallink{theorem}{http://planetphysics.us/encyclopedia/Formula.html}:
{\bf Theorem I.} The addition of vectors is {\bf commutative} and {\bf associative},
i.e. neither the order nor the grouping of the addends has an
influence on the sum of any number of vectors.
Thus, the fundamental laws of ordinary algebraic summation of
scalars hold good for vectors, without any reservation whatever.
If, in a chain-like arrangement of any number of vectors, the
end of the last coincides with the beginning of the first vector, then
the sum of all these vectors is \emph{nil}. Thus, in Fig. 5,
$${\bf A} + {\bf B} + {\bf C} + {\bf D} + {\bf E} = 0$$
\begin{figure}
\includegraphics[scale=.8]{Fig5.eps}
\end{figure}
A vector is \emph{nil} or \emph{zero}, ${\bf R} = 0$, if its tensor vanishes, $R = 0$. In
fact, this remark is scarcely necessary.
The sum of any number of vectors having the \emph{same direction}
(i.e. of vectors parallel and of the same sense) is a vector of the
same direction. In this particular case the tensor of the sum is
equal to the sum of the tensors. Thus, the common sum is a
particular case of the vector-sum.
Now, let us take the case of two or more equal vectors; then
we see at once that
$${\bf A} + {\bf A} = 2{\bf A}$$
is a vector of the same direction as ${\bf A}$ but of twice its tensor,
i.e. $2A$, and that analogous properties belong to $3{\bf A}$, $4{\bf A}$, and so
on. Again, understanding by $\frac{1}{2}{\bf A}$, $\frac{1}{3} {\bf A}$, etc., vectors which, repeated
$2$, $3$, etc., times (as addends), give the vector ${\bf A}$, and recurring to
the generally known limit-reasoning, we obtain the meaning of
$$n {\bf A}$$
where $n$ is any real \emph{positive}
scalar number, whole, fractional or
irrational. Thus, $n{\bf A}$ will be a vector which has the same
direction as ${\bf A}$ and the tensor of which is $nA$. In other terms,
$n{\bf A}$ will be the vector ${\bf A}$ stretched in the ratio $n:1$.
Thus, if ${bf \hat{a}}$ be a \emph{unit-vector} having the direction of ${\bf A}$, remembering
the definition of tensor, we may write
$${\bf A} = A{bf \hat{a}}$$
Any vector ${\bf A}$ may be represented in this way. Now $A$ is one
scalar, and ${bf \hat{a}}$ implies two scalars, for instance the angles $\theta$, $\epsilon$;
thus we see again that any vector implies $1+2 = 3$ scalars.
The addition of two (or more) vectors may be illustrated most
simply by regarding them as defining translations in space of, say,
a material particle. The translation ${\bf A}$ carries the particle from
$p$ to $p'$ (Fig. 6), the subsequent translation ${\bf B}$ carries it from $p'$ to $p''$.
\begin{figure}
\includegraphics[scale=.8]{Fig6.eps}
\end{figure}
The result of ${\bf A}$ followed by ${\bf B}$, or of ${\bf B}$ followed by ${\bf A}$, i.e. ${\bf A} + {\bf B}$
or ${\bf B} + {\bf A}$, is to carry the particle from $p$ to $p''$. Similarly, if ${\bf A}$, ${\bf B}$
be velocities of translation, ${\bf A} + {\bf B}$ will be the resultant velocity.
The same applies to angular velocities, to accelerations or forces.
If ${\bf A}$, ${\bf B}$ denote two forces acting simultaneously on a
material particle, ${\bf A} + {\bf B}$ will be the resultant force acting on that particle.
more to come soon...
\end{document}
