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| Coordinate
Geometry (Analytic Geometry) in Three-dimensional Space |
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Plane
in a three-dimensional (3D) coordinate system |
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Equations of
a plane in a coordinate space
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The intercept
form of the equation of a plane |
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A plane in a 3D
coordinate system, examples |
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Equation of the
plane through three points, example |
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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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| A
plane in
a
3D coordinate system,
examples |
| Example:
Find the equation of
the plane through three points, A(-1,
1, 4), B(2,
4, 3)
and C(-3,
4, 1).
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| Solution: |
| Equation of a plane
Ax
+ By
+ Cz
+ D = 0 is determined by
the components (the
direction cosines)
of the normal vector N
= Ai + Bj
+ Ck and
the coordinates of any point of the plane. |
| The normal vector is perpendicular to the plane determined by given
points, and as the vector |
| AB
= rB -
rA
= (2i + 4j + 3k)
-
(-i
+ j + 4k)
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rB
-
rA
= 3i + 3j -
k,
and the vector
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AC =
rC -
rA
= -2i
+ 3j -
3k
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| then,
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| By plugging the coordinates, of any given point, into equation
of the plane determines the value of the coefficient
D.
Thus, plugging |
| A(-1,
1, 4) into
-
6x
+ 11y + 15z + D = 0
gives -
6
· (-1)
+ 11
· 1
+ 15
· 1
+ D = 0, D = -77. |
| Therefore,
-
6x + 11y + 15z -
77 = 0 is the equation of the plane
the given points lie on. |
| The coordinate of the remaining two points
B
and C
must also satisfy the obtained equation, prove. |
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| Example:
Given are points,
A(-1,
1, 1)
and B(3,
-2,
6), find the equation of a plane which is normal to
the
vector
AB
and passes through the point A.
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| Solution:
According to the given conditions
the vector
AB
= N
so,
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| N
= rB -
rA
= (3i -
2j + 6k)
-
(-i
+ j + k) = 4i
-
3j + 5k
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| As the plane should pass through the point
A |
| plug
A(-1,
1, 1) into
4x -
3y + 5z + D = 0
=>
4 · (-1)
-
3 · 1
+ 5
· 1
+ D = 0,
D = 2 |
| Therefore,
the equation of the plane P
::
4x -
3y + 5z + 2 = 0. |
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| Example:
Segments that a plane cuts on the axes,
x
and y, are
l = -1 and
m = -2
respectively, find the standard or general equation of the plane if it passes through the point
A(3, 4, 6).
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| Solution:
As the given point A(3, 4,
6) and the segments, l =
-1 and
m = -2
must satisfy the intercept form of the equation of plane,
then
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| Coordinate
geometry contents |
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© 2004 - 2020, Nabla Ltd. All rights reserved. |
