Vectors In The Space - (10.2)
1. A Right-handed Coordinate System
We use a right-handed
!
x
,
y
,
z
"
coordinate system. A point in the space can be represented by
a
,
b
,
c
where
a
,
b
, and
c
are real numbers. The system consists of three axes:
x
!
axis,
y
!
axis, and
z
!
axis, three
planes:
xy
!
plane,
yz
!
plane, and
xz
!
plane, and eight octants. We can describe the planes and octants as
follows:
x
!
axis:
y
!
0,
z
!
0
y
!
axis:
x
!
0,
z
!
0
z
!
axis:
x
!
0,
y
!
0
,
xy
!
plane:
z
!
0
yz
!
plane:
x
!
0
xz
!
plane:
y
!
0
,
first octants:
x
"
0,
y
"
0, and
c
"
0
Example
Describe the plane which is parallel to xy
!
plane and has a point
!
!
1,2,3
"
.
Since it is parallel to
xy
!
plane, a point
!
a
,
b
,
c
"
on the plane has a constant
c
. Since the point
!
!
1,2,3
"
is in
the plane and
c
!
3, the third component of every point is 3. So, the equation of this plane is:
z
!
3.
-4
-2
2
4
z
-5
5
y
4
x
z
!
3
2. Distance Between Two Points
The
distance
between two points
P
!
x
1
,
y
1
,
z
1
"
and
Q
!
x
2
,
y