1. Vector addition is COMMUTATIVE:
a + b = b + a
2. Vector addition is ASSOCIATIVE:
a + (b + c) = (a + b) + c
3. There is a zero vector 0 such that:
a + 0 = a
4. Every vector a has a negative –a such that:
a + (–a) = 0
5. For every vector a we have:
1.a = a
6. If r is a scalar and a and b are vectors, then:
r(a + b) = ra + rb
7. If r and s are scalars and a is a vector, then:
(r + s)a = ra + sa
8. If r and s are scalars and a is a vector, then:
(rs)a = r(sa)
A vector space is any set V for which it is possible to
define a notion of addition (that is, a rule that combines two elements of the set V to produce a new element of V) and scalar multiplication (that is, a means
to multiply elements of V by numbers) so that the
above eight rules hold. Certainly the set of all twodimensional vectors forms a vector space, as does the
set of all three-dimensional vectors, but examples of
vector spaces need not be of this type. For example, let
V be the set of all functions from the set of all real
numbers to the set of real numbers. One can add any
two functions f and g:
(f + g)(x) = f(x) + g(x)
and multiply functions by numbers:
(rf)(x) = rf(x)
One checks that all eight rules above hold (here 0
is the function that takes the constant value zero), making V a vector space. Also, the set of all 3×3 matrices is
a vector space (one can add two matrices and multiply
matrices by scalars), as is the set of all COMPLEX NUMBERS. (One can add two complex numbers and one can
multiply complex numbers by real numbers.)
Properties of vectors and their algebraic manipulations have been studied extensively by scholars for centuries. That mathematicians have isolated the eight key
properties that make vectors work the way they do
allows one to immediately apply all that is known
about vectors to any system, no matter how abstract it
may be, that satisfies these eight basic rules. For example, mathematicians have proved that every vector space
must have a BASIS. Consequently, there must be a basis
for the set of all functions and for the set of all 3×3
matrices. Identifying one possible basis for the set of all
functions leads to a study of FOURIER SERIES.
See also GROUP; LINEARLY DEPENDENT AND INDEPENDENT.
velocity The study of motion examines three fundamental notions: distance, velocity, and acceleration.
The distance traveled by a moving object (also
called its displacement) is the total length of the path it
moved along. If the object travels along a straight-line
path, then its motion is said to be rectilinear. (Motion
that is not rectilinear is called curvilinear.) If, in addition to being rectilinear, an object travels equal distances D in equal periods of time T, then its motion is
said to be uniform. This is the easiest type of motion to
analyze. In this setting, the ratio
is constant and
is called the (uniform) velocity v of the object, v = ,
and the picture of a velocity-vs.-time graph is a horizontal line at height v. The formula D = v × T, coincidentally, is the equation for the area of the rectangle of
width T and height v. This shows, in this simple scenario at least, that displacement equals the area under
the velocity graph.
If, in rectilinear motion, the speed of an object
changes with time, as when a car accelerates from rest to
highway speed, then the analysis of velocity and displacement is more complicated. If f (t) denotes the position
D
–
T
D
–
T
522 velocity
a + b = b + a
2. Vector addition is ASSOCIATIVE:
a + (b + c) = (a + b) + c
3. There is a zero vector 0 such that:
a + 0 = a
4. Every vector a has a negative –a such that:
a + (–a) = 0
5. For every vector a we have:
1.a = a
6. If r is a scalar and a and b are vectors, then:
r(a + b) = ra + rb
7. If r and s are scalars and a is a vector, then:
(r + s)a = ra + sa
8. If r and s are scalars and a is a vector, then:
(rs)a = r(sa)
A vector space is any set V for which it is possible to
define a notion of addition (that is, a rule that combines two elements of the set V to produce a new element of V) and scalar multiplication (that is, a means
to multiply elements of V by numbers) so that the
above eight rules hold. Certainly the set of all twodimensional vectors forms a vector space, as does the
set of all three-dimensional vectors, but examples of
vector spaces need not be of this type. For example, let
V be the set of all functions from the set of all real
numbers to the set of real numbers. One can add any
two functions f and g:
(f + g)(x) = f(x) + g(x)
and multiply functions by numbers:
(rf)(x) = rf(x)
One checks that all eight rules above hold (here 0
is the function that takes the constant value zero), making V a vector space. Also, the set of all 3×3 matrices is
a vector space (one can add two matrices and multiply
matrices by scalars), as is the set of all COMPLEX NUMBERS. (One can add two complex numbers and one can
multiply complex numbers by real numbers.)
Properties of vectors and their algebraic manipulations have been studied extensively by scholars for centuries. That mathematicians have isolated the eight key
properties that make vectors work the way they do
allows one to immediately apply all that is known
about vectors to any system, no matter how abstract it
may be, that satisfies these eight basic rules. For example, mathematicians have proved that every vector space
must have a BASIS. Consequently, there must be a basis
for the set of all functions and for the set of all 3×3
matrices. Identifying one possible basis for the set of all
functions leads to a study of FOURIER SERIES.
See also GROUP; LINEARLY DEPENDENT AND INDEPENDENT.
velocity The study of motion examines three fundamental notions: distance, velocity, and acceleration.
The distance traveled by a moving object (also
called its displacement) is the total length of the path it
moved along. If the object travels along a straight-line
path, then its motion is said to be rectilinear. (Motion
that is not rectilinear is called curvilinear.) If, in addition to being rectilinear, an object travels equal distances D in equal periods of time T, then its motion is
said to be uniform. This is the easiest type of motion to
analyze. In this setting, the ratio
is constant and
is called the (uniform) velocity v of the object, v = ,
and the picture of a velocity-vs.-time graph is a horizontal line at height v. The formula D = v × T, coincidentally, is the equation for the area of the rectangle of
width T and height v. This shows, in this simple scenario at least, that displacement equals the area under
the velocity graph.
If, in rectilinear motion, the speed of an object
changes with time, as when a car accelerates from rest to
highway speed, then the analysis of velocity and displacement is more complicated. If f (t) denotes the position
D
–
T
D
–
T
522 velocity