Math 200, Spring
2010
Handout 1
Section 12-1
Vectors in the Plane
Section 12-2
Vectors in Three Dimensions
Three-Dimensional Rectangular Coordinate System
Coordinate axes
: The
x
-axis,
y
-axis,
z
-axis are perpendicular to each other.
Right-hand rule
: The fingers of the right hand curl from the positive
x
-axis to
the positive
y
-axis.
The thumb points in the positive
z
-direction.
Coordinates
of
point
P
are the ordered triple
( )
, ,
a b c
of real numbers.
Coordinate planes
in
3
ℝ
are defined by setting one of the coordinates to zero.
The
xy
-plane consists of the points
( )
, ,0
a b
and is defined by equation
0
z
=
.
The
yz
-plane consists of the points
( )
0, ,
b c
and is defined by equation
0
x
=
.
The
xz
-plane consists of the points
( )
,0,
a
c
and is defined by equation
0
y
=
.
The coordinate planes divide
3
ℝ
into eight
Octants.
The set of points
( )
, ,
a b c
with
, ,
0
a b c
>
is called the
First Octant
.
In two-dimensional geometry, the graph of an equation involving
x
and
y
is a
curve
in
2
ℝ
. In three-dimensional geometry, the graph
of an equation involving
x, y,
and
z
is a
surface
in
3
ℝ
.
1.
Sketch the points
( )
2,3,5
and
( )
1, 3,2
− −
on
2.
What does the equation
4
x
=
represent in
2
ℝ
?
a single set of coordinate axes.
What does the equation
4
x
=
represent in
3
ℝ
?
Illustrate with a sketch.