scalar In
VECTOR
analysis, any quantity that is a real
number and not a vector is called a scalar. For exam-
ple, the number 2 is the scalar coefficient of the vector
vin the vectorial expression 2v. The
DOT PRODUCT
of
two vectors (also called their scalar product) is a rule
for multiplying two vectors to yield a scalar result.
In physics, any number or measurement that does
not involve the concept of direction is called a scalar.
For example, length, mass, energy, and temperature are
scalar quantities. The notion of speed is also a scalar
quantity, but that of
VELOCITY
is not.
In
MATRIX
theory, the entries of a matrix are some-
times called scalars. A square matrix with all entries off
the main diagonal equal to zero and all entries on the
main diagonal equal in value is called a scalar matrix.
Such a matrix is a scalar multiple of the
IDENTITY
MATRIX
I.
scale Two geometric figures are said to be scaled ver-
sions of each other if it is possible to match the points
of one figure with points in the second in such a way
that, if Pand Qare any two points of the first figure
and P′and Q′are the corresponding points of the sec-
ond, then the ratio of lengths |PQ|: |P′Q′| is always
the same fixed positive value k. (The number kis called
the scale factor.)
SIMILAR FIGURES
are scaled figures.
Any enlarged or reduced figure produced by a modern
photocopier is a scaled version of the original figure.
If a geometric figure is enlarged by a scale factor
k, then all lengths in that picture increase (or decrease
if 0 < k< 1) by a factor of k. All
ANGLE
s in that figure
remain unchanged. Consequently, if an a×brectangle
is enlarged by a scale factor k, then the resultant fig-
ure remains a rectangle and has side-lengths ka and
kb. Consequently, the
AREA
of the figure has increased
by a factor k2. As the notion of area of a rectangle
defines our notion of area for all geometric figures
(such as a triangle, a polygon, or a circle) we have that
the areas of all figures increase, upon scaling, by the
factor k2. Similarly, since a rectangular prism of
dimensions a×b×cscales to become rectangular
prism of dimensions ka ×kb ×kc, volumes of all geo-
metric solids, upon scaling, increase by a factor of k3.
These observations have some interesting conse-
quences in the natural world. For instance, the amount
of heat loss a mammal experiences is proportional to
the amount of surface it has in contact with the air,
whereas the amount of food an animal must eat in
order to generate body warmth depends on the volume
of muscle it possesses. As surface area grows by a fac-
tor of k2and volume by a factor of k3, larger mammals
possess a smaller surface area per volume ratio than
smaller mammals. Thus large mammals (such as polar
bears) are better suited to arctic conditions than smaller
mammals (such as squirrels)—they lose less heat per
unit of body mass and require less food (in relation to
their mass) to maintain enough heat production. King
penguins in Antarctica are significantly larger than
their counterparts in other regions of the world. As
another example, the speed a fish can move is governed
by the volume of muscle it possesses. Large fish have
the advantage of having a smaller surface area to vol-
ume ratio than smaller fish. They not only have more
muscle power, but also experience less surface-area fric-
tion with the surrounding water (per unit of muscle)
than their smaller counterparts.
In mathematics, the study of scale plays an impor-
tant role in defining
DIMENSION
and is used to analyze
FRACTAL
s. In another context, the markings on the axes
of a
GRAPH OF A FUNCTION
constitute the “scale” of
the diagram. Changing the scale of the axes has the
equivalent effect of enlarging or reducing the graph
itself. The markings on a ruler are also called a scale,
and measuring the same object with two rulers of dif-
ferent scales is equivalent to the act of measuring two
scaled versions of the figure with a single ruler.
scatter diagram (scatter plot, Galton graph) If a sci-
entific study records numerical information about two
features of the individuals or events under examination
(such as height and shoe size of participating adults, or
temperature and pressure at which certain climatic
events occur), then that
DATA
can be represented in a
scatter diagram. Specifically, if the pair (xi,yi) represents
the two numerical facts recorded about the ith individ-
ual, then point (xi,yi) is plotted on C
ARTESIAN COORDI
-
NATES
. A scatter diagram is the resultant graph when
all points are displayed. The points are not joined by
lines.
A scatter diagram will indicate any relationship
between the xand yvariables. If the points seem to lie
near a straight line, they are said to be linearly corre-
lated. One can then perform a mathematical test to
determine the degree of correlation by calculating the
456 scalar