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| Coordinate
Geometry (Analytic Geometry) in Three-dimensional Space |
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Plane
in a three-dimensional (3D) coordinate system |
Equations of
a plane in a coordinate space
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The equation of a
plane in a 3D coordinate system |
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The Hessian normal form of the
equation of a plane
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The intercept
form of the equation of a plane |
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| Equations of a plane in a coordinate space
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| The equation of a
plane in
a
3D coordinate system |
| A plane in space is defined by three points (which don’t all lie on the same line) or by a point and a normal
vector to the plane. |
| Then, the scalar product of the vector
P1P
= r -
r1, drawn from the
given point P1(x1,
y1, z1) of the plane to any point
P(x,
y, z)
of the plane, and the normal vector N
= Ai + Bj
+ Ck, is zero, that is |
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which is called the vector equation of a
plane. |
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Or,
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| Therefore, |
| (x
-
x1)
· A
+ (y -
y1)
· B
+ (z -
z1)
· C
= 0 |
| giving |
A
· x
+ B · y
+ C · z
+ D = 0 |
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where, D
=
-
(Ax1+
By1+
Cz1) |
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the general equation of a plane in 3D
space. |
| If plane passes through the origin
O
of a coordinate system then its coordinates, x =
0, y
= 0, and z
= 0 plugged into the equation of the plane, give |
| A
· 0
+ B · 0
+ C · 0
+ D = 0 => D = 0. |
| Thus, the condition that plane passes through the
origin is D
= 0. |
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| The Hessian normal form of the
equation of a plane
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| Position of a plane in space can also be defined by the length of the normal, drawn from the origin to the plane
and by angles,
a, b and
g, that the normal forms with the
coordinate axes.
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| If
N°
is the unit vector of the normal then its components are, cosa,
cosb
and
cosg.
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| Then the projection of the position vector
r, of any point
P(x,
y, z) of the plane, onto the normal has the length
p. |
| Since the projection is determined by expression |
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| that is,
r
· N°
= p or
written in the coordinates |
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x
· cosa
+ y · cosb
+ z · cosg
-
p = 0 |
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| represents the
equation of a plane in the Hessian
normal
form. |
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| The intercept form of the equation of a
plane
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| If,
l,
m
and n
are the intercepts of
x,
y
and z
axes and a plane respectively, then projections of these segments in
direction of the normal drawn from the origin to the
plane are all equal to the length of the normal, that is
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| By plugging these values of cosines into Hessian normal
form of the equation of plane, obtained is
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| the intercept form of the equation of plane. |
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| Coordinate
geometry contents |
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