line in space

line in space

A line

l

in

ℝ

3

can be thought as the intersection of two planes and therefore can be represented by two simultaneous equations of planes of the form

{

A

1

⁢

x

+

B

1

⁢

y

+

C

1

⁢

z

+

D

1

=

0

A

2

⁢

x

+

B

2

⁢

y

+

C

2

⁢

z

+

D

2

=

0

,

(1)

where

A

i

,

B

i

,

C

i

,

D

i

are real and

A

i

2

+

B

i

2

+

C

i

2

>

0

.

Eliminating one of the variables

x

,

y

,

z

from the pair of equations, one obtains an equation of another plane. Thus, if one first eliminates

x

and then

y

, one may obtain simultaneous equations of the form

{

y

=

m

⁢

z

+

p

x

=

n

⁢

z

+

q

.

(2)

Obtaining equations exactly of the form of those in (2) from the equations in (1) may not be possible. For example, if

A

1

=

B

1

=

0

,  then the first equation in (1) simplifies to

C

1

⁢

z

+

D

1

=

0

.  Note though that, in this particular case, one can solve for

z

and plug this value into the equation

A

2

⁢

x

+

B

2

⁢

y

+

C

2

⁢

z

+

D

2

=

0

in to obtain the equation of a line on the

x

⁢

y

-plane. The of this entry deals with the case in which (2) is obtainable.

Note that an ordered triple

(

x

0

,

y

0

,

z

0

)

satisfies (1) if and only if it also satisfies (2), and hence the system of equations (2) represents the same line as well. Separately, both of the equations in (2) are

equations of planes

, of which the former is

parallel

to the

x

-axis (

x

can change independently of

y

and

z

) and the latter is parallel to the

y

-axis (

y

can change independently of

z

and

x

). The line itself is the

intersection

of these two planes, and it can be projected along the planes indicated by the individual equations in (2) onto the

y

⁢

z

-plane and the

z

⁢

x

-plane, respectively.

In a narrower meaning, the equation

y

=

m

⁢

z

+

p

represents the

projection

line of

l

onto the

y

⁢

z

-plane and the equation

x

=

n

⁢

z

+

q

the projection line of

l

onto the

z

⁢

x

-plane.

A

line in space

can be represented also without using planes. We can derive another way by starting with two vectors: one vector

r

→

0

drawn from the origin to a chosen point of the line (the so-called

position vector

of this point) and another vector (the so-called

direction vector

)

u

→

determining the

direction

of the line. Then there exists a real number

t

such that the position vector

r

→

of an arbitrary point of the line can be expressed as

r

→

=

r

→

0

+

t

⁢

u

→

.

(3)

This is the

vector-formed equation of the line

. Since every vector can be given by three

coordinates

of the end-point, we may write

r

→

=

(

x

y

z

)

,

r

→

0

=

(

x

0

y

0

z

0

)

,

u

→

=

(

a

b

c

)

.

Thus, (3) reads

(

x

y

z

)

=

(

x

0

y

0

z

0

)

+

t

⁢

(

a

b

c

)

,

and we obtain three scalar equations

{

x

=

x

0

+

t

⁢

a

y

=

y

0

+

t

⁢

b

z

=

z

0

+

t

⁢

c

.

(4)

These are the

parametric equations

of the line

(3), with the

parameter

t

taking all real values. The contents of (4) is often written with proportions:

x

-

x

0

a

=

y

-

y

0

b

=

z

-

z

0

c

(

=

t

)

References

1

L. Lindelöf

:

Analyyttisen geometrian oppikirja

.  Kolmas painos.  Suomalaisen Kirjallisuuden Seura, Helsinki (1924).

Title

line in space

Canonical name

LineInSpace

Date of creation

2013-03-22 17:28:52

Last modified on

2013-03-22 17:28:52

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

12

Author

pahio (2872)

Entry type

Topic

Classification

msc 53A04

Classification

msc 51N20

Synonym

line in

ℝ

3

Synonym

space line

Related topic

LineInThePlane

Related topic

DistanceOfNonParallelLines

Related topic

AngleBetweenTwoLines

Related topic

AnalyticGeometry

Defines

direction vector

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