Patterns of Relationship Between Variables
With measurement, observations become quantitative, and observed patterns or regularities take the form of mathemati-cal relationships. You should be familiar with the types of rela-tionships we consider here.Direct proportionality: Consider the numbers in the tables below. A B A B A B A A2 B 1 3 1 −5 1 3 1 1 3 2 6 2 −10 2 12 2 4 12 3 9 3 −15 3 27 3 9 27 4 12 4 −20 4 48 4 16 48
Tables i and ii exhibit relationships of the general type B = kA, or equivalently B/A = k, where k remains constant as B and A change. In i, k = 3; in ii, k = −5. In a relationship like this, we say that B varies directly as A, or that B is directly proportional to A. The word proportion means ratio. In direct proportionality, B doubles as A doubles, B triples as A triples, etc., and so forth, so that the ratio of the two values remains the same.
The relationship in iii is less obvious, but we can find a pat-tern by adding a column (highlighted) to (iii) listing the value of A2 for each value of A. We then see that each value in the B
col-umn is three times the corresponding value in the A2 column:
B = 3A2, or B/A2 = 3. In iii we can say that B is proportional to A2.
In Fig. 1a, B has been plotted against A for table i. The result is a straight line through the origin. A straight line is always equally steep. In Fig. 1a, whenever A increases horizontally by 1, B increases vertically by 3. It is much like a staircase (Fig. 2a) where if the treads all have the same width, the risers must all have the same height or else the steepness of the stairs will change. Figure 1b is a graph of table ii. The steepness is uniform
here too, but now whenever A increases by 1, B decreases verti-cally by 5. If k is negative, the graph read left to right is like a descending staircase (Fig. 2b). In contrast, the steepness of Fig-ure 1c, which plots B against A for table iii, is not uniform: The graph gets progressively steeper, like the staircase in Fig. 2c, in which the treads all have the same width but the risers become progressively higher.
The ratio of the riser to the tread 3/1 = 3 for Fig. 1a and Fig. 2a gives the steepness of the line, and is called the slope of the line. If you advance from a point (x1, y1) to a point (x2, y2)
along a graph of y vs. x (see Fig. 3), the x coordinate changes by an amount Δx = x2 − x1 and the y coordinate changes by
a corresponding amount Δy = y2 − y1 (Δ denotes “the change
in . . .”, so Δx means “the change in x”, and so forth). Then the slope m of this straight line is given by
Slope m y x y y x x = = − − Δ Δ 22 11 (1)
Mathematical Tools
i. ii. iii. iii redrawn.
(b) (c) (a) 12 0 1 2 3 1 2 3 –5 –10 –15 30 20 10 0 B 9 6 3 0 1 2 3 4 A A A B B
FiguRe 2 Staircase analogs of graphs in Fig. 1.
FiguRe 1 Graphs of relationships in tables i–iii. B versus A for (a) B = 3A (b) B = −5A (c) B = 3A2. 12 0 1 2 3 (1, –5) (3, 27) (2, 12) (1, 3) 1 2 3 (2, –10) (3, –15) (b)B = –5A (c)B = 3A2 (a)B = 3A –5 –10 –15 30 20 10 0 B 9 6 3 0 1 2 (1, 3) (2, 6) (3, 9) (4, 12) 3 4 A A A B B y y2 y1 (x1, y1) (x2, y2) Dy Dx Dy = y2– y1 Dx = x2– x1 x1 x2 x q
FiguRe 3 Graph of a straight line. As you advance from a point (x1, y1) to a point (x2, y2), your x coordinate changes by an amount Δx = x2 − x1 and your y coordinate correspondingly changes by an amount Δy = y2 −y1.
Any two points (x1, y1) and (x2, y2) along the straight line
will give you the same slope, precisely because a straight line is always equally steep. In contrast, different choices of (x1, y1) and (x2, y2) in Fig. 1c will not give you the same value of m; straight lines connecting pairs of points along this graph do not have equal steepness.
If we read left to right, Δx = x2 −x1 is always positive. Then
the slope Δy/Δx is negative when Δy = y2 − y1 is negative.
A negative slope tells us the vertically plotted values are going down and the graph looks like a descending staircase. If the graph is horizontal, then the size Δy of the “riser” is zero, and thus the slope Δy/Δx is also zero.
In the standard x and y notation, x really means “whatever variable you have plotted horizontally” and y really means “whatever variable you have plotted vertically”. Thus, for table (i) and Fig. 1a, x becomes A, y becomes B, and the slope becomes
m B A B B A A = = − − Δ Δ 2 1 2 1
. If we let (A1, B1) be the origin (0, 0), and let (A2, B2) be any other point (A, B) along the line, m B
A B A B A = = − − = Δ Δ 0 0 .
Any corresponding pair of values from table i gives B/A = 3, the proportionality constant we found previously.
When two variables A and B are directly proportional (B = kA ), the graph of B (vertically) vs. A (horizontally) is a straight line through the origin, and that the proportionality constant k and the slope m are one and the same thing.
This provides a basis for interpreting the slope in different situations. For example, suppose a car’s clock and odometer are set to zero at the start of a trip. As the car travels at constant speed, subsequent readings are as follows
Clock Reading Odometer Reading (km)
0 0
1 80
2 160
3 240
4 320
Its speed is given by Speed=time spent travelingdistance traveled . Since the speed is constant, we can use any pair of values:
Speed=80 = = = = 1 160 2 240 3 320 4 80 km h km h km h km h km h If we plot the values as in Fig. 4, the slope m y x = Δ Δ here becomes Δ Δ ( ) ( ) odometer reading
stopwatch reading . Since we have plotted quan-tities with units, we must retain those units in the calculation of the slope. We can calculate the slope from any two points. Here it is convenient to let the origin (0,0) be one of the points:
m= − − = − − = = = 80 0 1 0 160 0 2 0 80 km h km h etc km h .... ....
In this case, calculating the slope gives us the speed. We can interpret the slope of either graph as the speed.
The slope is the same in both cases, even though one graph looks steeper than the other, i.e. the two graphs do not make the same angle with the horizontal. It is only meaningful to compare the steepness of two graphs that have both the same horizontal scale and the same vertical scale.
Let’s generalize. Speed is a rate of change; it tells you the rate at which the distance changes with respect to time. The word per means “in each”; 80 km/h or 80 km per hour tells you that the car advances 80 km in each hour of driving time. The slope of a graph is also a rate. It tells you how many units you go up (or down) for each unit that you go across (how many units of riser for each unit of tread):
1. The slope always tells you the rate of change of the vertically plotted variable as the horizontal variable increases.
2. The units of the slope are units of the vertical variable per unit of the horizontal variable.
Since the constant of proportionality k in B = kA gives the slope of the graph of B vs. A, it likewise can always be inter-preted as a rate.
Consider another example. Suppose you are required to “weigh” a large sand pile. You have at your disposal a scale cali-brated in kg and a pail. You use the scale to find the mass of the pail. You then fill the pail with sand, find its mass, and subtract off the mass of the empty pail to find the mass of a “standard pailful” of sand. Now you need only determine the number of times you can fill the pail in the same way, and multiply the mass of the standard pailful by this number to get the total mass of the pile:
Total mass = (mass of a standard pailfuil)
× (number of pailfuls) (2)
The pail can hold a fixed volume of sand. The “number of pailfuls”, like the number of cubic meters or liters in other situa-tions, is the number of units of volume, which ordinarily we just
320 240 160 80 0 320 240 160 80 0 1 2 3
Clock reading (hours) Distance read by odometer (km)
Distance read by odometer (km)
Clock reading (hours) (a)
(b) 4
1 2 3 4
FiguRe 4 Graph of the data for the traveling car. The graph is shown here for two different choices of scale. Both are correct; the choice of scale is arbitrary.
call “the volume”. The “mass of a standard pailful” is the mass in each (or per) unit of volume. We can thus rewrite Eq. 2 as
Total mass = (Mass per unit of voulme) × (Volume) The mass per unit of volume is the constant rate in this sit-uation. The total mass is proportional to the total volume of the sand pile, but the mass of each pailful (to within experimental error) is the same, and is a property of the sand itself. This prop-erty is called the density (ρ) of the sand, which is given by
Mass = Density × Volume or m = rV or ρρ =m V This is like B = kA or k B
A
= ; density is now the proportion-ality constant. It is a rate because it tells how much mass of a substance you get for each unit of volume.
Proportionality Between Po wers of Variables
The graph of B vs. A for table iii is not a straight line. However, since in iii B is proportional to A2, a graph of B vs. A2 is a straightline, as Figure A-5 shows. Note carefully that the numerical val-ues from the A2column and not those from the A column, are
equally spaced along the horizontal axis. Since x and y are sim-ply stand-ins for whatever variables we plot horizontally and vertically, the slope is in general given by
Slope variable plotted vertically
variable plotted hor = Δ
Δ
( )
( iizontally) (3)
where each “Δ” is a latter value of the variable minus an earlier value
In this case, the slope (m = k = 3) is Δ Δ
B A2
( )
. We can in general have a direct proportionality between a power of one variable and a power (the same or different) of another.A A A -1==1 B A A A −−2 2 1 = B 1 2 2 20 1 1 5 1 1 10 2 1 9= .0 11 1.25 2 0.5 5 3 1 9= .0 11 0.55 iv v
Inverse and inverse square relationships: These powers can also be negative. In tables iv and v A and B are not propor-tional, but the values in the second and third columns are: In iv, B = kA−1 and k = 10; in (v), B = kA−2 and k = 5. We can rewrite
these as iv B = k/A and v B = k/A2. In (iv), B is said to be inversely
proportional to A. The relationship between B and A in v is called an inverse square relationship. In both of these relation-ships, B decreases as A increases. Fig. 6a shows this for v. But in v, B is proportional to 1/A2, so a plot of B vs 1/A2 is a straight
line (Fig. 6b).
We can rewrite the relationship in v as A2B = k. For the
given values, k = 5. If we look at the pairs of values of A and B in v 12× 5 = 5), 22× 1.25 = 5, and 32× 0.55 = 5. More generally, we
can write A B k A B12 1= , 22 2=k, and so forth.
Linear but not proportional: the initial value problem. When-ever two quantities are directly proportional, a graph of one against the other is a straight line, or linear. Does it necessarily follow that when a relationship between two quantities is linear, the quantities are proportional? Consider the following.
The trip graphed in Fig. 4 is repeated at the same constant speed as before. The car clock is set at zero where it previously started the trip, but this time the driver has been visiting with a friend 10 km up the road, and zeroed her odometer when she FiguRe 5 Graph of B versus A2 for table iii. For B = 3 A2, the graph of B versus A2 is a straight line, although the graph ofB versus A is not. Note that the evenly spaced values along the horizontal axis are values of A2, not A.
(1, 3) 1 (A = 1) (A = 2) (A = 3) 0 3 12 27 (9, 27) B (4, 12) 4 9 A2 5.0 2.5 0 5.0 2.5 0 1 2 (a) (b) (0.25, 1.25) (1.5) (2,1.25) (3,0.55) (0.11,0.55) (1.5) B A A2 1 B 3 0.25 0.50 0.75 1.00
FiguRe 6 Two graphs of the inverse square relationship B = 5/ A2. (a) Plotting B against A shows that B decreases as A increases.
(b) Plotting B against 1/A2 shows that B is proportional to 1/A2. Note that the values of 1/A2 must be placed in ascending order left to right along the horizontal axis.
was there. As the following table shows, her time and trip dis-tance (first two columns) remain proportional, but to each trip distance, we must add 10 km to get her odometer reading.
The odometer readings are not proportional to the clock readings: if we double the clock reading from 1 h to 2 h, the odometer reading goes from 90 km to 170 km; it does not dou-ble. Nevertheless, both graphs in Fig. 7 are straight lines.
Note that the graph of trip distance passes through the origin; the graph of odometer reading does not. At t = 0, the odometer reading is 10 km, the initial distance from her friend’s house when the driver starts her trip. At each subsequent value of t, the odometer reading exceeds the trip distance by this same 10 km. The value at t = 0, where the graph crosses the vertical axis, is called the vertical or y intercept. The vertical intercept always repre-sents an initial value. The two graphs are the same, except that one builds from an initial distance value of zero, and the other from an initial distance value of 10 km. However, two linearly related variables are proportional only when the vertical intercept—the initial value of the vertically plotted variable—is zero.
The relationship between the values in the first and third columns above, which is graphed in Fig. 7, can be expressed as
Odometer reading = Initial odometer reading + Speed × Time spent Traveling This relationship between the vertically and horizontally plotted variables holds true whenever the rate of change (in this instance, the speed) is constant, so that in general
Vertically plotted variable = Initial value of vertically
plotted variable + Rate of plotted variable
× Horizontally plotted variable (4a)
But the rate of change is the slope of the corresponding straight line graph, so we can write Eq. 4a symbolically as
y = yo + mx (4b)
Equation 4b is a general form of the equation of a straight line, called the slope-intercept form. The initial value yo is
the vertical intercept, the value of y at x = 0. It is commonly denoted by b in mathematics texts (so that Eq. 4b appears as y = mx + b). More importantly, Eq. 4a is the interpretation of A-4b that gives it meaning when it is used to represent real world situations.
A direct proportionality relationship always has the form One variable = Constant × Other variable
Whenever the slope is constant, the relationship Δy = m Δx has this form, so that changes in y are proportional to changes in x. But unless the vertical intercept yo (or b) is zero, the
equa-tion for a straight line, y = yo + mx, does not have this form, and
y itself is not proportional to x. 1 0 10 80 90 160 170 240 250 320 330 2
Clock reading (hours) Distance read by odometer (km)
3 4
FiguRe 7 More data for the traveling car. Distances from the original starting place are in black; distances from the friend’s house are in red.
Components of Vectors
a a a a a a a a a x y x y y x = = = + = cos sin tan φ φ φ 2 2 / b b b b x y = < = > cos ( ) sin ( ) φ φ 0 0 O ax a ay x y f b by x y O bx fVectors
a a a a x y = = sin cos sin sin θ φ θ φ a a a ax ay az = = + + cosθ 2 2 2 tan cos φ θ = = a a a a y x z / /
unit Vectors
Two-dimensional Cartesian: a=axi+ayjThree dimensional Cartesian:
a=axi+ayj+azk Two-dimensional polar: a=ar ruˆ +aφ φuˆ z x az a ax xy projection of a ay y q f ax ay y x a i^ j^ x ax az z ay a y j^ i^ k^ a f uf af ^ ur ar ^
Adding Vectors
s a b= + = + = + sx ax bx sy ay by a b b a (commutative law) d (e f) (d e) f (associat + = + + + = + + iive law) d a b a= − = + −b = − = − ( ) dx ax bx dy ay byMultiplication of Vectors
Multiplication of a vector by a scalar: b a | | = = = = c b ca b ca b c a x x y y
Dot product (or scalar product) of two vectors: a b a b b a . . . = = = =
abcosφ a b( cos )φ b a( cos )φ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ i i j j k k 1 i j i k j k . . . . . . = = = = = = 0 a b a a . . = + + = = + + a b a b a b a a a a x x y y z z x y z 2 2 2 2 a b s = a + b b d d + e+ f d+e e +f a a b e f a+b b+a –b d = a – b b a 1.4a a –0.5a a b f b cos f a cos f
Cross product (or vector product) of two vectors: c a b |c| |a b| = × = × = absinφ
Direction of c is perpendicular to the plane of a and b, determined by the right-hand rule.
b a a b i i j j k k i j k j × = − × × = × = × = × = × ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 0 kk i k i j=ˆ ˆ ˆ ˆ× = a b× =(a b a by z− z y) ˆ (i+ a b a bz x− x z)ˆ (j+a b a bx y− y xx x y z x y z a a a b b b )ˆ ˆ ˆ ˆ k i j k = a b c a b a c ( a) b a b a b a s × + = × + × × = × = × = ( ) ( ) ( ) ( ) ( ) ( s s s s ccalar). a b c b c a c a b a b c a c b a . . . . . ( ) ( ) ( ) ( ) ( ) ( × = × = × × × = − bb c) b a c = a× b f
