Math2011, 3Dimensional Space; Vectors
1.1
Rectangular Coordinates
Points in the space are presented by their coordinates, similar to the case for
points in a plane. Take three lines in the space which are mutually perpendicular
and we call them the xaxis, yaxis and the zaxis.
The intersection of these
three lines is called the origin
O
.
The xyplane is the plane containing both the xaxis and the yaxis, and its
similar for the yzplane and the xzplane.
The first (x), second (y), third (z) coordinates of a point
P
are the perpen
dicular distances from
P
to the yzplane, xzplane, xyplane respectively.
For a function in one variable
f
, the graph of
f
is the set of points (
x, y
) in
the plane satisfying
y
=
f
(
x
). The graph is a geometric object closely related
to the function
f
.
In general, if
f
is a function in two variables. The graph of
f
is the set of
points (
x, y, z
) in the space satisfying
z
=
f
(
x, y
). It is a surface in the space
and is closely related to the study of
f
.
If (
a, b, c
) is a point in the space. The distance from the origin to this point
is
√
a
2
+
b
2
+
c
2
. This result is done by applying the Pythagoras theorem for
twice. More generally, the distance between the points (
a, b, c
) and (
α, β, γ
) in
the space is
p
(
a

α
)
2
+ (
b

β
)
2
+ (
c

γ
)
2
.
Example 1.1
The set of points
(
x, y, z
)
in the space satisfying
x
2
+
y
2
+
z
2
= 4
is the set of points having a distance
2
from the origin. Thus, it is a sphere with
center the origin and radius
2
.
Example 1.2
The set of points
(
x, y, z
)
in the space satisfying
y
2
+
z
2
= 4
is
the set of points having a distance
2
from the xaxis. Thus, it is a cylinder with
its axis the xaxis and radius
2
.
1.2
Vectors
Definition 1.3
A vector is an ordered collection of numbers.
There are some operations defined for vectors:
Definition 1.4
Addition:
(
x
1
, ..., x
n
) + (
y
1
, ..., y
n
) = (
x
1
+
y
1
, ..., x
n
+
y
n
)
1
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Definition 1.5
Scalar multiplication:
a
(
x
1
, ..., x
n
) = (
ax
1
, ...ax
n
)
Theorem 1.6
For all vectors
x
,
y
,
z
and for all numbers
a
,
b
,
1.
(
a
+
b
)
x
=
ax
+
bx
2.
(
ab
)
x
=
a
(
bx
)
3.
a
(
x
+
y
) =
ax
+
ay
4.
x
+
y
=
y
+
x
5.
(
x
+
y
) +
z
=
x
+ (
y
+
z
)
Definition 1.7
Inner product (Dot product):
(
x
1
, ..., x
n
)
·
(
y
1
, ..., y
n
) =
x
1
y
1
+
...
+
x
n
y
n
Theorem 1.8
For all vectors
x
,
y
,
z
, and for all numbers
a
,
1.
(
ax
)
·
y
=
a
(
x
·
y
)
2.
(
x
+
y
)
·
z
=
x
·
z
+
y
·
z
3.
x
·
y
=
y
·
x
Vectors and their usual operations can be interpreted geometrically in the
following way:
1. A vector (
x, y, z
) is represented by a point in the space whose coordinates
are
x
,
y
and
z
. It can also be represented by an arrow pointing from the
origin to the point with coordinates
x
,
y
,
z
.
2. The addition of two vectors
u
+
v
is the forth vertex of the parallelogram
whose three other vertices are
u
,
O
and
v
.
3. The scalar multiplication by a number
a
is to lengthen vectors by a factor
a
.
Definition 1.9
The length of a vector
x
is the nonnegative number
√
x
·
x
. It
is also denoted by

x

.
Remark 1.10
The length of a vector

x

=
√
x
·
x
is the distance from the
origin to the point representing
x
.
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