equation of plane

equation of plane

The position of a plane

τ

can be fixed by giving the

position vector

O

⁢

Q

→

of the

projection

point

Q

of the origin on the plane.

Let the length of the position vector be

r

and the angles formed by the vector with the positive

coordinate

axes

α

,

β

,

γ

. Let

P

=

(

x

,

y

,

z

)

be an arbitrary point. Then

P

is in the plane

τ

iff its projection on the line

O

⁢

Q

coincides with

Q

, i.e. iff (

http://planetmath.org/Iff

) the projection of the

coordinate way

of

P

is

r

. This may be expressed as the equation

x

⁢

cos

⁡

α

+

y

⁢

cos

⁡

β

+

z

⁢

cos

⁡

γ

=

r

or

x

⁢

cos

⁡

α

+

y

⁢

cos

⁡

β

+

z

⁢

cos

⁡

γ

-

r

=

0

,

(1)

which thus is the equation of the plane.

Conversely, we may show that a first-degree equation

A

⁢

x

+

B

⁢

y

+

C

⁢

z

+

D

=

0

(2)

between the

variables

x

,

y

,

z

represents always a plane. In fact, we may without hurting generality suppose that

D

≦

0

. Now

R

:=

A

2

+

B

2

+

C

2

>

0

.  Thus the length of the

radius vector

(

http://planetmath.org/PositionVector

) of the point

(

A

,

B

,

C

)

is

R

. Let the angles formed by the radius vector with the positive coordinate axes be

α

,

β

,

γ

. Then we can write

A

=

R

⁢

cos

⁡

α

,

B

=

R

⁢

cos

⁡

β

,

C

=

R

⁢

cos

⁡

γ

(cf.

direction cosines

). Dividing (2) termwise by

R

gives us

x

⁢

cos

⁡

α

+

y

⁢

cos

⁡

β

+

z

⁢

cos

⁡

γ

+

D

R

=

0

,

where

D

R

≦

0

. The last equation represents a plane whose

distance

from the origin is

-

D

R

and whose

normal line

forms the angles

α

,

β

,

γ

with the coordinate axes.

Since the

coefficients

A

,

B

,

C

are proportional to the direction cosines of the

normal vector

of this plane, they are direction numbers of the normal line of the plane.

Examples.

The equations of the

coordinate planes

are

x

=

0

(

y

⁢

z

-plane),

y

=

0

(

z

⁢

x

-plane),

z

=

0

(

x

⁢

y

-plane);

the equation of the plane through the points

(

1

,

0

,

0

)

,

(

0

,

1

,

0

)

and

(

0

,

0

,

1

)

is

x

+

y

+

z

=

1

.

The plane can be represented also in a vectoral form, by using the position vector

r

→

0

of a point of the plane and two

linearly independent

vectors

u

→

and

v

→

parallel

to the plane:

r

→

=

r

→

0

+

s

⁢

u

→

+

t

⁢

v

→

.

(3)

Here,

r

→

means the position vector of arbitrary point of the plane,

s

and

t

are real parameters. In the coordinate form, (3) may be e.g.

{

x

=

x

0

+

s

⁢

a

+

t

⁢

d

,

y

=

y

0

+

s

⁢

b

+

t

⁢

e

,

z

=

z

0

+

s

⁢

c

+

t

⁢

f

.

Title

equation of plane

Canonical name

EquationOfPlane

Date of creation

2013-03-22 17:28:48

Last modified on

2013-03-22 17:28:48

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

13

Author

pahio (2872)

Entry type

Topic

Classification

msc 51N20

Related topic

DirectionCosines

Related topic

SurfaceNormal

Related topic

RuledSurface

Related topic

AnalyticGeometry

Related topic

AngleBetweenLineAndPlane

Related topic

IntersectionOfQuadraticSurfaceAndPlane

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