**
Plane
**

*
General equation of plane.
Normal vector.
*

*
Equation of plane in segments on
axes.
*

*
Equation of plane going through the given point
*

*
and perpendicular to the given vector.
*

*
Parametric equation of plane.
*

*
Parallelism condition of planes.
*

*
Perpendicularity condition of
planes.
*

*
Distance between two points
*
.

*
Distance from point to plane.
*

*
Distance between parallel planes.
*

*
Angle between planes.
*

*
*

**
A general equation of plane:
**

*
Àõ +
Âó
+
Ñ
z
+
D
=
*
0 ,

where
*
À
*
,
*
B
*
and
*
C
*
aren't equal to zero simultaneously.

Coefficients
*
À
*
,
*
B
*
and
*
C
*
are coordinates of
**
normal vector
**
of the
plane (
i.e. vector, perpendicular to the
plane
).

At
*
*
*
À
*
0,
*
Â
*
0,
*
Ñ
*
0 and
*
D
*
*
*
0
we receive
**
an
equation of plane in segments
**

*on**:*

**axes**

where
*
a
*
= –
*
D
*
/
*
A
*
,
*
b
*
= –
*
D
*
/
*
B
*
,
*
c
*
= –
*
D
*
/
*
C
*
. This plane
goes through the points (
*
a
*
,
0, 0
), (
0,
*
b
*
, 0
) and (
0, 0,
*
ñ
*
), i.e. it cuts off segments
*
a
*
,
*
b
*
and
*
c
*
long on the coordinate axes.

**
An equation of plane
**
,
going through a point
(

*õ*

_{ 0 },

*ó*

_{ 0 }

**,**

*z*

_{ 0 }) and perpendicular to a vector (

*À*,

*Â , C*) :

*
À
*
(
*
õ
*
–
*
õ
*
_{
0
}
*
*
)
*
+ Â
*
(
*
ó
*
–
*
ó
*
_{
0
}
*
*
)
*
+ Ñ
*
(
*
z
*
–
*
*
*
z
*
_{
0
}
*
*
)
*
=
*
0 .

**
**

**
A parametric equation of plane
**
, passing through a point (

*õ*

_{ 0 },

*ó*

_{ 0 }

**,**

*z*

_{ 0 }) and two noncollinear vectors (

*a*

_{ 1 },

*b*

_{ 1 }

**,**

*c*

_{ 1 })

_{ }and

**(**

*a*

_{ 2 }

*,*

*b*

_{ 2 }

*,*

*c*

_{ 2 }) , set in a rectangular cartesian system of soordinates:

**
A parallelism condition of
planes
**

*Àõ + Âó + Ñ z+ D =*0 and

*E*

*õ + F ó + Gz+ H =*0 :

*
AF
*
–
*
BE
*
=
*
BG
*
–
*
CF
*
=
*
AG
*
–
*
CE
*
= 0 .

**
A perpendicularity condition
of planes
**

*Àõ + Âó + Ñ z+ D =*0 and

*E*

*õ + F ó + Gz+ H =*0 :

*
À
E
+
Â
F
+
Ñ
G
=
*
0 .

**
A
distance between two points
**
(

*x*

_{ 1 }

*,*

*y*

_{ 1 },

*z*

_{ 1 }

*)*

_{ }and

**(**

*x*

_{ 2 }

*,*

*y*

_{ 2 }

*,*

*z*

_{ 2 }) :

**
A distance from a point
**
(

*õ*

_{ 0 },

*ó*

_{ 0 }

**,**

*z*

_{ 0 })

*to a plane**Àõ + Âó + Ñ z + D =*0 :

**
A distance between parallel planes
**

*Aõ + By + Cz + D*= 0 and

*Aõ + By + Cz + Å*= 0

**
An angle
**

*between planes**Àõ + Âó + Ñ z+ D =*0 and

*E*

*õ + F ó + Gz+ H =*0 :