specified, but we are informed that it must be a whole
number between 1 and 10 inclusive. The value of the
variable yalso is not specified, but we are told that its
value depends on the chosen value for n. For this reason,
yis called a dependent variable: once a value for nhas
been declared, a value for ythen follows. The variable n
is called an independent variable because its value is not
hinged to the values of any other variables.
Two or more variables are also said to be indepen-
dent if the value chosen of one variable has no effect on
the possible value of the second. For example, if xrep-
resents the price of a pound of sugar on any given day,
and ythe number of students taking math at college,
then xand yare independent variables. If each math
student were to buy a pound of sugar, then the total
money spent, z= xy, is a new variable. Here, zis not
independent of xor y.
In algebra, symbols are also often used to denote
quantities whose values are assumed known. For exam-
ple, in the expression ax + b= 0, it is assumed that aand
bare specified numbers, and that xis the value to be
determined. (We have x= – .) Here we are following a
convention, first established by R
ENÉ
D
ESCARTES
(1596–1650), stating that lowercase letters at the end of
the alphabet (such as x, y, and z) are to represent
unknown quantities, and lowercase letters from the
beginning of the alphabet (such as a, b, and c) known
quantities. (The letters from the middle of the alphabet
must be interpreted via context.)
See also
CONSTANT
;
FUNCTION
;
HISTORY OF EQUA
-
TIONS AND ALGEBRA
(essay).
variance See
STATISTICS
:
DESCRIPTIVE
.
vector In physics and engineering it is often appropri-
ate to describe physical quantities, such as velocity or
force, in terms of both magnitude and direction. For
example, wind speed is completely specified by a single
number that specifies its magnitude (wind strength),
but wind velocity refers to both the speed of the wind
and the direction in which the air is moving. Gravita-
tional force acts on objects on the Earth’s surface with
a magnitude (strength) dependent on the mass of the
object and in a direction pulling the object toward the
center of the Earth. Any quantity that is specified by
both a magnitude and a direction is called a vector.
Vectors are usually depicted as line segments with
arrows indicating the direction of the vector. The
length of the line segment indicates the magnitude of
the vector. For example, a planar vector of magnitude 2
in the direction east is drawn in the plane as a horizon-
tal line segment, 2 units long, with an arrow pointing
to the right. The location at which this line segment
starts is not important. All line segments 2 units long
pointing to the right represent the same vector. Thus
one is free to translate a vector to any starting position
in the plane, as long as the length of the vector and its
direction remain unchanged.
A vector is usually denoted in textbooks by bold-
face letters. For example, the symbol amight represent
one vector and banother. However, in some texts, and
often in handwritten notes, it is customary to underline
letters or use overbars or arrows to indicate that the
quantities represented are vectorial: a,–
a, or →
a. The
magnitude or length of a vector ais denoted |a| (or
sometimes as
a
).
A vector can be specified by an ordered pair of
numbers. For example, the vector of magnitude 2 in
the direction east can be represented as the pair: a=
<2,0>. This vector “carries particles” 2 units to the
right, and no units up or down. The vector b= <1,1>
“carries particles” 1 unit right and 1 unit up. This rep-
resents the vector of magnitude √
–
2 (the length of the
diagonal of the unit square) in the direction northeast.
The vector c= <–1,–1> is the same vector but pointing
in the opposite direction. The vector 0= <0,0> is called
the zero vector and represents a quantity with no mag-
nitude or direction.
The use of angle brackets to denote vectors is com-
mon in physics and engineering, although some texts
use parentheses to represent vectors. This can be con-
fusing, for parentheses are also used to denote points in
the plane.
If P= (p1, p2) and Q= (q1, q2) represent two points
in the plane, then a vector that “carries a particle” from
position Pto Qis given by: <q1– p1,q2– p2>. This vec-
tor is sometimes denoted
→
PQ. The point Pis called the
initial point of the vector, and Qis the terminal point.
(However, keep in mind that a vector is not fixed at any
specific location. Any vector of the same length and
direction as
→
PQ represents the same vector, even though
the initial and terminal points might be different.)
If a= < a1,a2> is a vector, then the numbers a1and
a2are called the components of the vector. The length
b
–
a
vector 519