5.1
Chapter Five
More Dimensions
5.1 The Space
R
n
We are now prepared to move on to spaces of dimension greater than three.
These
spaces are a straightforward generalization of our Euclidean space of three dimensions.
Let
n
be a positive integer.
The
n-dimensional Euclidean space R
n
is simply the set of
all ordered
n-
tuples of real numbers
x
=
(
,
,
,
)
x
x
x
n
1
2
K
. Thus
R
1
is simply the real
numbers,
R
2
is the plane, and
R
3
is Euclidean three-space.
These ordered
n-
tuples are
called
points
, or
vectors
. This definition does not contradict our previous definition of a
vector in case
n
=3 in that we identified each vector with an ordered triple (
,
,
)
x
x
x
1
2
3
and
spoke of the triple as being a vector.
We now define various arithmetic operations on
R
n
in the obvious way.
If we
have vectors
x
=
(
,
,
,
)
x
x
x
n
1
2
K
and
y
=
(
,
,
,
)
y
y
y
n
1
2
K
in
R
n
,
the sum
x
y
+
is defined
by
x
y
+
=
+
+
+
(
,
,
,
)
x
y
x
y
x
y
n
n
1
1
2
2
K
,
and multiplication of the vector
x
by a scalar
a
is defined by
a
ax
ax
ax
n
x
=
(
,
,
,
)
1
2
K
.
It is easy to verify that
a
a
a
(
)
x
y
x
y
+
=
+
and (
)
a
b
a
b
+
=
+
x
x
x
.
Again we see that these definitions are entirely consistent with what we have done
in dimensions 1, 2, and 3-there is nothing to unlearn.
Continuing, we define the
length
,
or
norm
of a vector
x
in the obvious manner
| |
x
=
+
+
+
x
x
x
n
1
2
2
2
2
K
.
The
scalar product
of
x
and
y
is