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
+ Ñz
+
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+
Ñz+
D = 0 and Eõ+
Fó+
Gz+ H =
0 :
AF
– BE = BG – CF = AG – CE = 0 .
A perpendicularity condition
of planes
Àõ+
Âó+
Ñz+
D
=
0 and Eõ+
Fó+
Gz+ H
=
0
:
ÀE+
ÂF+
ÑG
= 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
+
Ñz
+ D =
0 :
A distance between parallel planes Aõ + By + Cz + D = 0 and Aõ + By + Cz + Å = 0
An angle
between planes
Àõ+
Âó+
Ñz+
D =
0
and Eõ+
Fó+
Gz+ H =
0 :
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