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Unformatted text preview: 11 Vectors 111 Symmetry in physics In this chapter we introduce a subject that is technically known in physics as
symmetry in physical law. The word “symmetry” is used here with a special
meaning, and therefore needs to be deﬁned. When is a thing symmetrical—how
can we deﬁne it? When we have a picture that is symmetrical, one side is somehow
the same as the other side. Professor Hermann Weyl has given this deﬁnition of
symmetry: a thing is symmetrical if one can subject it to a certain operation and
it appears exactly the same after the operation. For instance, if we look at a vase
that is leftandright symmetrical, then turn it 180° around the vertical axis, it
looks the same. We shall adopt the deﬁnition of symmetry in Weyl’s more general
form, and in that form we shall discuss symmetry of physical laws. Suppose we build a complex machine in a certain place, with a lot of compli
cated interactions, and balls bouncing around with forces between them, and so on.
Now suppose we build exactly the same kind of equipment at some other place,
matching part by part, with the same dimensions and the same orientation, every
thing the same only displaced laterally by some distance. Then, if we start the
two machines in the same initial circumstances, in exact correspondence, we ask:
will one machine behave exactly the same as the other? Will it follow all the mo
tions in exact parallelism? Of course the answer may well be no, because if we
choose the wrong place for our machine it might be inside a wall and interferences
from the wall would make the machine not work. All of our ideas in physics require a certain amount of common sense in their
application; they are not purely mathematical or abstract ideas. We have to under
stand what we mean when we say that the phenomena are the same when we move
the apparatus to a new position. We mean that we move everything that we
believe is relevant; if the phenomenon is not the same, we suggest that something
relevant has not been moved, and we proceed to look for it. If we never ﬁnd it,
then we claim that the laws of physics do not have this symmetry. On the other
hand, we may ﬁnd it—we expect to ﬁnd it—if the laws of physics do have this
symmetry; looking around, we may discover, for instance, that the wall is pushing
on the apparatus. The basic question is, if we deﬁne things well enough, if all the
essential forces are included inside the apparatus, if all the relevant parts are moved
from one place to another, will the laws be the same? Will the machinery work
the same way? It is clear that what we want to do is to move all the equipment and essential
inﬂuences, but not everything in the world—planets, stars, and all—for if we do
that, we have the same phenomenon again for the trivial reason that we are right
back where we started. No, we cannot move everything. But it turns out in
practice that with a certain amount of intelligence about what to move, the ma
chinery will work. In other words, if we do not go inside a wall, if we know the
origin of the outside forces, and arrange that those are moved too, then the ma
chinery will work the same in one location as in another. 112 Translations We shall limit our analysis to just mechanics, for which we now have sufﬁcient
knowledge. In previous chapters we have seen that the laws of mechanics can be
summarized by a set of three equations for each particle: m(d2x/d12) = F,, m(d2y/dt2) = F,, m(d22/dt2) = F,. (11.1) 11—1 11—1 Symmetry in physics 11—2 Translations 11—3 Rotations 11—4 Vectors 11—5 Vector algebra 11—6 Newton’s laws in vector notation 11—7 Scalar product of vectors Fig. 11—1. systems. Two parallel coordinate Now this means that there exists a way to measure x, y, and 2 on three perpendicu
lar axes, and the forces along those directions, such that these laws are true.
These must be measured from some origin, but where do we put the origin? All
that Newton would tell us at ﬁrst is that there is some place that we can measure
from, perhaps the center of the universe, such that these laws are correct. But we
can show immediately that we can never ﬁnd the center, because if we use some
other origin it would make no diﬂerence. In other words, suppose that there are
two people—Joe, who has an origin in one place, and Moe, who has a parallel
system whose origin is somewhere else (Fig. 11—1). Now when Joe measures the
location of the point in space, he ﬁnds it at x, y, and 2 (we shall usually leave 2 out
because it is too confusing to draw in a picture). Moe, on the other hand, when
measuring the same point, will obtain a dlfferent x (in order to distinguish it, we
will call it x’), and in principle a different y, although in our example they are
numerically equal. So we have I I x = x — a, y = y, z’ = 2. (11.2) Now in order to complete our analysis we must know what Moe would obtain for
the forces. The force is supposed to act along some line, and by the force in the
xdirection we mean the part of the total which is 1n the xdirection, which is
the magnitude of the force times this cosine of its angle with the xaxis. Now we
see that Moe would use exactly the same prOJection as Joe would use, so we have
a set of equations F, = F,, F, = F,, F, = F,. (11.3) These would be the relationships between quantities as seen by Joe and Moe. The question is, if Joe knows Newton’s laws, and if Moe tries to write down
Newton’s laws, will they also be correct for him? Does it make any diﬂ‘erence from
which origin we measure the points? In other words, assuming that equations
(11.1) are true, and the Eqs. (11.2) and (11.3) give the relationship of the measure
ments, is it or is it not true that (a) m(d2x’/dtz) = (b) m(d2y’/dt2) = Fy’;
(c) m(d22’/dt2) = 12,? (11.4) In order to test these equations we shall differentiate the formula for x’
twice. First of all dx’_d( _ )_E_@.
“‘“X a ‘d: dt Now we shall assume that Moe’s origin is ﬁxed (not moving) relative to Joe’s;
therefore a is a constant and da/dt = 0, so we ﬁnd that dx’/dt = dx/dt’
and therefore
a’2x’/alt2 = d2x/dt2; therefore we know that Eq. (11.4a) becomes
m(d2x/dt2) = F,“ (We also suppose that the masses measured by Joe and Moe are equal.) Thus the
acceleration times the mass is the same as the other fellow’s. We have also found
the formula for F5, for, substituting from Eq. (11.1), we ﬁnd that F,» = F1. Therefore the laws as seen by Moe appear the same; he can write Newton’s
laws too, with different coordinates, and they will still be right. That means that 11—2 there is no unique way to deﬁne the origin of the world, because the laws will
appear the same, from whatever position they are observed. This is also true: if there is a piece of equipment in one place with a certain
kind of machinery in it, the same equipment in another place will behave in the
same way. Why? Because one machine, when analyzed by Moe, has exactly the
same equations as the other one, analyzed by Joe. Since the equations are the same,
the phenomena appear the same. So the proof that an apparatus in a new position
behaves the same as it did in the old position is the same as the proof that the
equations when displaced in space reproduce themselves. Therefore we say that
the laws of physics are symmetrical for translational displacements, symmetrical
in the sense that the laws do not change when we make a translation of our co
ordinates. Of course it is quite obvious intuitively that this is true, but it is inter
esting and entertaining to discuss the mathematics of it. 11—3 Rotations The above is the ﬁrst of a series of ever more complicated propositions con
cerning the symmetry of a physical law. The next proposition is that it should
make no difference in which direction we choose the axes. In other words, if we
build a piece of equipment in some place and watch it operate, and nearby we
build the same kind of apparatus but put it up on an angle, will it operate in the
same way? Obviously it will not if it is a Grandfather clock, for example! If a
pendulum clock stands upright, it works ﬁne, but if it is tilted the pendulum falls
against the side of the case and nothing happens. The theorem is then false in
the case of the pendulum clock, unless we include the earth, which is pulling on
the pendulum. Therefore we can make a prediction about pendulum clocks if
we believe in the symmetry of physical law for rotation: something else is involved
in the operation of a pendulum clock besides the machinery of the clock, something
outside it that we should look for. We may also predict that pendulum clocks will
not work the same way when located in different places relative to this mysterious
source of asymmetry, perhaps the earth. Indeed, we know that a pendulum clock
up in an artiﬁcial satellite, for example, would not tick either, because there is no
effective force, and on Mars it would go at a different rate. Pendulum clocks do
involve something more than just the machinery inside, they involve something
on the outside. Once we recognize this factor, we see that we must turn the earth
along with the apparatus. Of course we do not have to worry about that, it is easy
to do; one simply waits a moment or two and the earth turns; then the pendulum
clock ticks again in the new position the same as it did before. While we are
rotating in space our angles are always changing, absolutely; this change does not
seem to bother us very much, for in the new position we seem to be in the same
condition as in the old. This has a certain tendency to confuse one, because it is
true that in the new turned position the laws are the same as in the unturned
position, but it is not true that as we turn a thing it follows the same laws as it does
when we are not turning it. If we perform sufﬁciently delicate experiments, we
can tell that the earth is rotating, but not that it had rotated. In other words, we
cannot locate its angular position, but we can tell that it is changing. Now we may discuss the effects of angular orientation upon physical laws.
Let us ﬁnd out whether the same game with Joe and Moe works again. This time,
to avoid needless complication, we shall suppose that Joe and Moe use the same
origin (we have already shown that the axes can be moved by translation to another
place). Assume that Moe’s axes have rotated relative to Joe’s by an angle 0.
The two coordinate systems are shown in Fig. 11—2, which is restricted to two
dimensions. Consider any point P having coordinates (x, y) in Joe’s system and
(x’, y’) in Moe’s system. We shall begin, as in the previous case, by expressing
the coordinates x’ and y’ in terms of x, y, and 6. To do so, we ﬁrst drop perpendic
ulars from P to all four axes and draw AB perpendicular to PQ. Inspection of the
ﬁgure shows that x’ can be written as the sum of two lengths along the x’axis,
and y’ as the difference of two lengths along AB. All these lengths are expressed 11—3 Fig. 11—2. Two coordinate systems
having different angular orientations. Fig. 11—3. Components of a force in
the two systems. in terms of x, y, and 0 in equations (11.5), to which we have added an equation
for the third dimension. x = xcos0 + ysin 0, y = ycos6 — xsin 9, (11.5) 2:2. The next step is to analyze the relationship of forces as seen by the two observers,
following the same general method as before. Let us assume that a force F, which
has already been analyzed as having components FE and F, (as seen by Joe), is
acting on a particle of mass m, located at point P in Fig. 11—2. For simplicity, let
us m0ve both sets of axes so that the origin is at P, as shown in Fig. 11—3. Moe
sees the components of F along his axes as Fri and F,,:. F, has components along
both the x’ and y’axes, and F1, likewise has components along both these axes.
To express F11 in terms of F, and Fy, we sum these components along the x’axis,
and in a like manner we can express Fy: in terms of F: and Fy. The results are F,: = F, cos 0 + F” sin 6,
F”: = F, cos 0 — F, sin 6,
F,» = F2. (11.6) It is interesting to note an accident of sorts, which is of extreme importance: the
formulas (11.5) and (11.6), for coordinates of P and components of F, respectively,
are of identical form. As before, Newton’s laws are assumed to be true in Joe’s system, and are
expressed by equations (11.1). The question, again, is whether Moe can apply
Newton’s laws—will the results be correct for his system of rotated axes? In other
words, if we assume that Eqs. (11.5) and (11.6) give the relationship of the measure
ments, is it true or not true that m(d2x’/dt2) = m(d2y’/dt2) = F),
m(d22’/dt2) = F2]? (11.7) To test these equations, we calculate the left and right sides independently, and
compare the results. To calculate the left sides, we multiply equations (11.5) by m,
and differentiate twice with respect to time, assuming the angle 0 to be constant.
This gives m(d2x’/a't2) = m(d2x/dt2) cos 0 + m(d2y/dt2) sin 6, m(d2y’/dt2) = m(d2y/d12) cos 0 — m(d2x/dt2) sin 0,
m(d2z’/dt2) = m(dzz/dt2). (11.8) We calculate the right sides of equations (11.7) by substituting equations (11.1)
into equations (11.6). This gives Fxr = m(d2x/dt2) cos 0 + m(d2y/dt2) sin 0,
Fyr = m(d2y/dt2) cos 0 — m(d2x/dt2) sin 0,
F, = m(d22/dt2). (11.9) Behold! The right sides of Eqs. (11.8) and (11.9) are identical, so we conclude
that if Newton’s laws are correct on one set of axes, they are also valid on any
other set of axes. This result, which has now been established for both translation
and rotation of axes, has certain consequences: ﬁrst, no one can claim his particular
axes are unique, but of course they can be more convenient for certain particular
problems. For example, it is handy to have gravity along one axis, but this is not
physically necessary. Second, it means that any piece of equipment which is
completely selfcontained, with all the forcegenerating equipment completely in
side the apparatus, would work the same when turned at an angle. 11—4 11—4 Vectors Not only Newton’s laws, but also the other laws of physics, so far as we know
today, have the two properties which we call invariance (or symmetry) under
translation of axes and rotation of axes. These properties are so important that a
mathematical technique has been developed to take advantage of them in writing
and using physical laws. The foregoing analysis involved considerable tedious mathematical work.
To reduce the details to a minimum in the analysis of such questions, a very power
ful mathematical machinery has been devised. This system, called vector analysis,
supplies the title of this chapter; strictly speaking, however, this is a chapter on
the symmetry of physical laws. By the methods of the preceding analysis we were
able to do everything required for obtaining the results that we sought, but in
practice we should like to do things more easily and rapidly, so we employ the
vector technique. We began by noting some characteristics of two kinds of quantities that are
important in physics. (Actually there are more than two, but let us start out with
two.) One of them, like the number of potatoes in a sack, we call an ordinary
quantity, or an undirected quantity, or a scalar. Temperature is an example of
such a quantity. Other quantities that are important in physics do have direction,
for instance velocity: we have to keep track of which way a body is going, not just
its speed. Momentum and force also have direction, as does displacement: when
someone steps from one place to another in space, we can keep track of how far
he went, but if we wish also to know where he went, we have to specify a direction. All quantities that have a direction, like a step in space, are called vectors. A vector is three numbers. In order to represent a step in space, say from the
origin to some particular point P whose location is (x, y, 2), we really need three
numbers, but we are going to invent a single mathematical symbol, r, which is
unlike any other mathematical symbols we have so far used.* It is not a single
number, it represents three numbers: x, y, and 2. It means three numbers, but
not really only those three numbers, because if we were to use a different coordinate
system, the three numbers would be changed to x’, y’, and 2’. However, we want
to keep our mathematics simple and so we are going to use the same mark to repre
sent the three numbers (x, y, z) and the three numbers (x’, y’, 2’). That is, we use
the same mark to represent the ﬁrst set of three numbers for one coordinate system,
but the second set of three numbers if we are using the other coordinate system.
This has the advantage that when we change the coordmate system, we do not
have to change the letters of our equations. If we write an equation in terms of
x, y, z, and then use another system, we have to change to x’, y’, 2’, but we shall
just write 1, with the convention that it represents (x, y, 2) if we use one set of axes,
or (x’, y’, 2’) if we use another set of axes, and so on. The three numbers which
describe the quantity in a g1ven coordinate system are called the components of the
vector in the direction of the coordinate axes of that system. That is, we use the
same symbol for the three letters that correspond to the same object, as seen from
different axes. The very fact that we can say “the same object” implies a physical
intuition about the reality of a step in space, that is independent of the components
in terms of which we measure it. So the symbol r will represent the same thing
no matter how we turn the axes. Now suppose there is another directed physical quantity, any other quantity,
which also has three numbers associated with it, like force, and these three
numbers change to three other numbers by a certain mathematical rule, if we
change the axes. It must be the same rule that changes (x, y, 2) into (x’, y’, 2’). In
other words, any physical quantity associated with three numbers which transform
as do the components of a step in space is a vector. An equation like F=r would thus be true in any coordinate system if it were true in one. This equation, * In type, vectors are represented by boldface; 1n handwritten form an arrow is used :72
11—5 of course, stands for the three equations F2: = x3 F1] = y; Fz = z:
or, alternatively, for
F2, = x’, F,,: y’, F, = z’. The fact that a physical relationship can be expressed as a vector equation assures
us the relationship is unchanged by a mere rotation of the coordinate system.
That is the reason why vectors are so useful in physics. Now let us examine some of the properties of vectors. As examples of vectors
we may mention velocity, momentum, force, and acceleration. For many purposes
it is convenient to represent a vector quantity by an arrow that indicates the direc
tion in which it is acting. Why can we represent force, say, by an arrow? Because
it has the same mathematical transformation properties as a “step in space.” We
thus represent it in a diagram as if it were a step, using a scale such that one unit
of force, or one newton, corresponds to a certain convenient length. Once we
have done this, all forces can be represented as lengths, because an equation like F=kr, where k is some constant, is a perfectly legitimate equation. Thus we can always
represent forces by lines, which is very convenient, because once we have drawn
the line we no longer need the axes. Of course, we can quickly calculate the three
components as they change upon turning the axes, because that is just a geometric
problem. 11—5 Vector algebra Now we must describe the laws, or rules, for combining vectors in various
ways. The ﬁrst such combination is the addition of two vectors: suppose that
a is a vector which in some particular coordinate system has the three components
(ax, ay, a,), and that b is another vector which has the three components (bx, by, b,).
Now let us invent three new numbers (ax + b,, a, + by, a, + b,). Do these form
a vector? “Well,” we might say, “they are three numbers, and every three numbers
form a vector.” No, not every three numbers form a vector! In order for it to be a
vector, not only must there be three numbers, but these must be associated with a
coordinate system in such a way that if we turn the coordinate system, the three
numbers “revolve” on each other, get “mixed up” in each other, by the precise
laws we have already described. So the question is, if we now rotate the coordinate
system so that (a1, a,,, az) become (arr, ayl, azr) and (b,, by, b,) become (by, by', by),
What do (a; + bx, ay + by, aZ + b,) become? Do they become (axr + by,
up + by, a, + by) or not? The answer is, of course, yes, because the prototype
transformations of Eq. (11.5) constitute what we call a linear transformation.
If we apply those transformations to a, and b1 to get a1: + by, we ﬁnd that
the transformed a, + b: is indeed the same as arr + by. When a and b are
“added together” in this sense, they wi11,form a vector which we may call c. We
would write this as c = a + b.
Now c has the interesting property
c = b + a,
as we can immediately see from its components. Thus also,
a+(b+c)= (a+b)+C. We can add vectors in any order.
What is the geometric signiﬁcance of a + b? Suppose that a and b were
represented by lines on a piece of paper, What would c look like? This is shown in 11—6 Fig. 11—4. We see that we can add the components of b to those of a most con
veniently if we place the rectangle representing the components of b next to that
representing the components of a in the manner indicated. Since b Just “ﬁts”
into its rectangle, as does a into its rectangle, this is the same as putting the “tail”
of b on the “head” of a, the arrow from the “tail” of a to the “head” of b being
the vector c. Of course, if we added a to b the other way around, we would put the
“tail” of a on the “head” of b, and by the geometrical properties of parallelograms
we would get the same result for c. Note that vectors can be added in this way
without reference to any coordinate axes. Suppose we multiply a vector by a number a, what does this mean? We
deﬁne it to mean a new vector whose components are aux, any, and 01:1,. We leave
it as a problem for the student to prove that it is a vector. Now let us consider vector subtraction. We may deﬁne subtraction in the
same way as addition, but instead of adding, we subtract the components. Or
we might deﬁne subtraction by deﬁning a negative vector, —b = — lb, and then
we would add the components. It comes to the same thing. The result is shown
in Fig. 11—5. This ﬁgure shows 11 = a — b = a + (—b); we also note that the
difference a — b can be found very easily from a and b by us1ng the equivalent
relation a = b + 11. Thus the difference 1s even easier to ﬁnd than the sum: we
just draw the vector from b to a, to get a — b! Next we discuss velocity. Why is velocity a vector? If position is given by the
three coordinates (x, y, 2), what is the velocity? The velocity is given by dx/dt,
dy/dt, and dz/dt Is that a vector, or not? We can ﬁnd out by differentiating the
expressions in Eq. (11.5) to ﬁnd out whether dx’/dt transforms in the right way.
We see that the components dx/dt and dy/dt d0 transform according to the same
law as x and y, and therefore the time derivative is a vector. So the velocity is a
vector. We can write the velocity in an interesting way as v = dr/dt. What the velocity is, and why 1t is a vector, can also be understood more pictorially:
How far does a particle move in a short time At? Answer: Ar, so if a particle is
“here” at one instant and “there” at another instant, then the vector difference
of the positions Ar = r2 — r1, which is in the direCIion of motion shown in Fig.
11—6, divided by the time interval At = t2 — 21, is the “average velocity” vector.
In other words, by vector velocity we mean the limit, as At goes to 0, of the
difference between the radius vectors at the time t + Ar and the time t, divided by A1:
v = lim (Ar/At) = dr/dt. All“) (11.10) Thus velocity is a vector because it is the difference of two vectors. It is also the
right deﬁnition of velocity because its components are dx/dt, dy/dt, and dz/dt.
In fact, we see from this argument that if we differentiate any vector with respect
to time we produce a new vector. So we have several ways of producing new
vectors: (1) multiply by a constant, (2) differentiate with respect to time, (3) add
or subtract two vectors. 11—6 Newton’s laws in vector notation In order to write Newton’s laws in vector form, we have to go just one step
further, and deﬁne the acceleration vector. This is the time derivative of the velocity
vector, and it is easy to demonstrate that its components are the second derivatives
of x, y, and z with respect to z: dv d dr d2r
a_d_t_ (25X?!) _3?§. (11.11)
d2), de do, d2y d1), (122
Z=_:_, =._=_, ,=—=—. 11.12
a dt at:2 a” dt dt2 0 dt alt2 ( ) 11—7 The addition of vectors. Fig. 1 1—4. 3:3—5 Fig. 11—5. The subtraction of vectors. Fig. 11—6. The displacement of a
particle in a short time interval At =
12 — f]. Fig. 1 1—7. A curved troiectory. Fig. 11—8. Diagram for calculating
the acceleration. With this deﬁnition, then, Newton’s laws can be written in this way: ma = F (11.13) or m(d2r/dtz) = F. (11.14) Now the problem of proving the invariance of Newton’s laws under rotation
of coordinates is this: prove that a is a vector; this we have just done. Prove that F
is a vector; we suppose it is. So if force is a vector, then, since we know acceleration
is a vector, Eq. (11.13) will look the same in any coordinate system. Writing it in
a form which does not explicitly contain x’s, y’s, and 2’s has the advantage that
from now on we need not write three laws every time we write Newton‘s equations
or other laws of physics. We write what looks like one law, but really, of course,
it is the three laws for any particular set of axes, because any vector equation
involves the statement that each of the components is equal. The fact that the acceleration is the rate of change of the vector velocity helps
us to calculate the acceleration in some rather complicated circumstances. Suppose,
for instance, that a particle is moving on some complicated curve (Fig. 11—7) and
that, at a given instant I, it had a certain velocity V], but that when we go to another
instant 12 a little later, it has a different velocity v2. What is the acceleration?
Answer: Acceleration is the difference in the velocity divided by the small time
interval, so we need the difference of the two velocities. How do we get the differ
ence of the velocities? To subtract two vectors, we put the vector across the ends
of v2 and v1; that is, we draw A as the difference of the two vectors, right? No!
That only works when the tails of the vectors are in the same place! It has no mean
in g if we move the vector somewhere else and then draw a line across, so watch out!
We have to draw a new diagram to subtract the vectors. In Fig. 11—8, v1 and v2
are both drawn parallel and equal to their counterparts in Fig. 11—7, and now we
can discuss the acceleration. Of course the acceleration is simply Av/At. It is
interesting to note that we can compose the velocity difference out of two parts;
we can think of acceleration as having two components, Av in the direction tangent
to the path and Av L at right angles to the path, as indicated in Fig. 11—8. The
acceleration tangent to the path is, of course, just the change in the length of the
vector, i.e., the change in the speed v: a” = dv/dt. (11.15) The other component of acceleration, at right angles to the curve, is easy to cal
culate, using Figs. 11—7 and 11—8. In the short time At let the change in angle
between V, and v2 be the small angle A6. If the magnitude of the velocity is called
v, then of course Av, = 1) A0
and the acceleration a will be (I, = v (AB/At). Now we need to know A0/At, which can be found this way: If, at the given moment,
the curve is approximated as a circle of a certain radius R, then in a time At the
distance s is, of course, 2) At, where v is the speed. A0 = v (At/R), or Ae/At = v/R. Therefore, we ﬁnd
a = vz/R’ as we have seen before. 11—7 Scalar product of vectors Now let us examine a little further the properties of vectors. It 15 easy to see
that the length of a step in space would be the same in any coordinate system.
That is, if a particular step 1' is represented by x, y, z, in one coordinate system, 11—8 and by x’, y’, z’ in another coordinate system, surely the distance r = lrl would
be the same in both. Now r=\/x2+yz+z2 So what we wish to verify is that these two quantities are equal. 1t is much more
convenient not to bother to take the square root, so let us talk about the square of
the distance; that is, let us ﬁnd out whether and also x2 + y2 + Z2 : x’2 + y’2 + z’2_ It had better be—and if we substitute Eq. (11.5) we do indeed ﬁnd that it is. So
we see that there are other kinds of equations which are true for any two coordinate
systems. Something new is involved. We can produce a new quantity, a function of
x, y, and 2, called a scalar function, a quantity which has no direction but which is
the same in both systems. Out of a vector we can make a scalar. We have to ﬁnd
a general rule for that. It is clear what the rule is for the case just considered:
add the squares of the components. Let us now deﬁne a new thing, which we call
a  a. This is not a vector, but a scalar; it is a number that is the same in all coordi nate systems, and it is deﬁned to be the sum of the squares of the three components
of the vector: aa = a3 + 113+ a3. (11.18) Now you say, “But with what axes?” It does not depend on the axes, the answer
is the same in every set of axes. So we have a new kind of quantity, a new invariant or scalar produced by one vector “squared.” If we now deﬁne the following
quantity for any two vectors a and b: ab = axbx + (1,1), + a217,, (11.19) we ﬁnd that this quantity, calculated in the primed and unprimed systems, also
stays the same. To prove it we note that it is true of a  a, b  b, and c c, where
c = a + b. Therefore the sum of the squares (a1 + b.,)2 + (a, + by)2 +
(a3 + b2)2 will be invariant: (ax + bag + (a1 + by? + (a. + b.)2 = (015' + 12.02
+ (011' + by’)2 + (“2’ ‘1' 17302. If both sides of this equation are expanded, there will be cross products of just the
type appearing in Eq. (11.19), as well as the sums of squares of the components
of a and b. The invariance of terms of the form of Eq. (11.18) then leaves the cross
product terms (11.19) invariant also. The quantity a  b is called the scalar product of two vectors, a and b, and it
has many interesting and useful properties. For instance, it is easily proved that a(b+c)=a'b+ac. (11.21) Also, there is a simple geometrical way to calculate a  b, without having to cal
culate the components of a and b: a  b is the product of the length of a and the
length of b times the cosine of the angle between them. Why? Suppose that we
choose a special coordinate system in which the xaxis lies along a; in those cir
cumstances, the only component of a that will be there is (1,, which is of course
the whole length of 3. Thus Eq. (11.19) reduces to a i b = arbz for this case,
and this is the length of a times the component of b in the direction of a, that is,
b cos 0: ab = abcosa. Therefore, in that special coordinate system, we have proved that a  b is the
11—9 L ngth of :1 times the length of b times cos 0. But if it is true in one coordinate system,
it is true in all, because a ~ b is independent of the coordinate system; that is our
argument. What good is the dot product? Are there any cases in physics where we need
it? Yes, we need it all the time. For instance, in Chapter 4 the kinetic energy was
called émvz, but if the objeét is movmg in space it should be the velocity squared
in the xdirection, the ydirection, and the zdirection, and so the formula for
kinetic energy according to vector analysis is K.E. = %m(vv) = %m(v§ + v: + 113). (11.22) Energy does not have direction. Momentum has direction; it is a vector, and it is
the mass times the velocity vector. Another example of a dot product is the work done by a force when something
is pushed from one place to the other. We have not yet deﬁned work, but it is
equivalent to the energy change, the weights lifted, when a force F acts through a distance 5:
Work = Fs (11.23) It is sometimes very convenient to talk about the component of a vector in a
certain direction (say the vertical direction because that is the direction of gravity).
For such purposes, it is useful to invent what we call a unit vector in the direction
that we want to study. By a unit vector we mean one whose dot product with
itself is equal to unity. Let us call this unit vector i; then i  i = 1. Then, if we want
the component of some vector in the direction of i, we see that the dot product
a  i will be a cos 0, i.e., the component of a in the direction of i. This is a nice
way to get the component; in fact, it permits us to get all the components and to
write a rather amusing formula. Suppose that in a given system of coordinates,
x, y, and 2, we invent three vectors: i, a unit vector in the direction x; j, a unit vector
in the direction y; and k, a unit vector in the direction 2. Note ﬁrst that i ‘ i = 1. What is i  j? When two vectors are at right angles, their dot product is zero.
Thus ii =1
ij =0 jj =1
ik=0 jk=0 k~k=1 (11.24) Now with these deﬁnitions, any vector whatsoever can be written this way:
a = axi + ayj + azk. (11.25) By this means we can go from the components of a vector to the vector itself.
This discussion of vectors is by no means complete. However, rather than
try to go more deeply into the subject now, we shall ﬁrst learn to use in physical
situations some of the ideas so far discussed. Then, when we have properly mastered
this basic material, we shall ﬁnd it easier to penetrate more deeply into the subject
without getting too confused. We shall later ﬁnd that it is useful to deﬁne another
kind of product of two vectors, called the vector product, and written as a X b.
However, we shall undertake a discussion of such matters in a later chapter. 1110 ...
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This note was uploaded on 06/18/2009 for the course PHYSICS none taught by Professor Leekinohara during the Spring '09 term at Uni. Nottingham  Malaysia.
 Spring '09
 LeeKinohara
 Physics

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