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Why is there unique plane which passes through given point and is parallel to given line

Find an equation for the plane that contains the following line and passes through point PEquation of a plane passing through a line and a separate point.Find plane which parallel to two vectors $L_1 ( 3,1,10)$ and $L_2(1,-1,1)$ passes through a point $M(7,-10,3)$Find the equation of the plane that passes through the line of intersection of the planes…Line equation through point, parallel to plane and intersecting lineEquation of a plane given one point and two planesEquation of plane passing through intersection of line and planeEquation of plane through a point and perpendicular to planesFind the Point Through which the Variable Line PassesGiven a plane and a point, find a line that passes through the point and is parallel to the plane

1

$begingroup$

I was trying to solve one question which is asking to find a plane which passes through given point and is parallel to given line.

The given point is $M(2,-5,3)$ and the given line is given as an interesection of the planes $2x-y+3z-1=0 text and 5x+4y-z-7=0$

It is still unclear for me why there is only one unique plane which can be answer, I think that there are more possible planes that can be answers to this.

analytic-geometry

share|cite|improve this question

edited Mar 13 at 20:30

someone123123

asked Mar 13 at 9:51

someone123123someone123123

445414

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Perhaps you might include the specific problem that you’re asking about.
  $endgroup$
  – amd
  Mar 13 at 19:40
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I inserted the given point and the line into the post
  $endgroup$
  – someone123123
  Mar 13 at 20:30
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Are you sure that the problem said for the plane to be parallel to that line? If it must instead include the line, then the solution is be unique.
  $endgroup$
  – amd
  Mar 13 at 20:35
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Yes, the question is asking about plane which is parallel to the given line.
  $endgroup$
  – someone123123
  Mar 13 at 20:45
  

  
  
  
  

add a comment | 

1

$begingroup$

I was trying to solve one question which is asking to find a plane which passes through given point and is parallel to given line.

The given point is $M(2,-5,3)$ and the given line is given as an interesection of the planes $2x-y+3z-1=0 text and 5x+4y-z-7=0$

It is still unclear for me why there is only one unique plane which can be answer, I think that there are more possible planes that can be answers to this.

analytic-geometry

share|cite|improve this question

edited Mar 13 at 20:30

someone123123

asked Mar 13 at 9:51

someone123123someone123123

445414

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Perhaps you might include the specific problem that you’re asking about.
  $endgroup$
  – amd
  Mar 13 at 19:40
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I inserted the given point and the line into the post
  $endgroup$
  – someone123123
  Mar 13 at 20:30
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Are you sure that the problem said for the plane to be parallel to that line? If it must instead include the line, then the solution is be unique.
  $endgroup$
  – amd
  Mar 13 at 20:35
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Yes, the question is asking about plane which is parallel to the given line.
  $endgroup$
  – someone123123
  Mar 13 at 20:45
  

  
  
  
  

add a comment | 

$begingroup$

I was trying to solve one question which is asking to find a plane which passes through given point and is parallel to given line.

The given point is $M(2,-5,3)$ and the given line is given as an interesection of the planes $2x-y+3z-1=0 text and 5x+4y-z-7=0$

It is still unclear for me why there is only one unique plane which can be answer, I think that there are more possible planes that can be answers to this.

analytic-geometry

share|cite|improve this question

edited Mar 13 at 20:30

someone123123

asked Mar 13 at 9:51

someone123123someone123123

445414

$endgroup$

share|cite|improve this question

edited Mar 13 at 20:30

someone123123

asked Mar 13 at 9:51

someone123123someone123123

445414

share|cite|improve this question

edited Mar 13 at 20:30

someone123123

asked Mar 13 at 9:51

someone123123someone123123

445414

asked Mar 13 at 9:51

someone123123someone123123

445414

asked Mar 13 at 9:51

someone123123someone123123

445414

3 Answers
3

active

oldest

votes

3

$begingroup$

Your are right, such plane is not unique. For example the planes $2x-y+3z=18$ and $5x+4y-z=-13$ pass through the point $(2,-5,3)$ and they are parallel to the given line.

More generally, through the given point, there is a unique line parallel to the given line, but then any plane through this second line is parallel to the given line.

share|cite|improve this answer

edited Mar 14 at 4:25

answered Mar 13 at 9:55

Robert ZRobert Z

101k1069142

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I'm thinking like this, but my teacher gave some explanation that I didn't really understand. She says that there is only one such plane
  $endgroup$
  – someone123123
  Mar 13 at 9:57
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Maybe the problem was "find a plane which passes through a given point and it is orthogonal to a given line.
  $endgroup$
  – Robert Z
  Mar 13 at 10:00
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Or perhaps the plane is supposed to include the intersection line of the two given planes (see updated question). Impossible to know without seeing the teacher’s solution.
  $endgroup$
  – amd
  Mar 14 at 0:24
  

  
  
  
  

add a comment | 

1

$begingroup$

You are right: there are infinitely mane planes passing through a point and parallel to a given line.

share|cite|improve this answer

answered Mar 13 at 9:55

AlessioDVAlessioDV

955114

$endgroup$

add a comment | 

1

$begingroup$

As another answer points out, the claim is false. Given a set of planes parallel to each other as well as to the given line, only one of those planes will pass through the given point.

share|cite|improve this answer

edited Mar 13 at 10:06

answered Mar 13 at 10:00

Oscar LanziOscar Lanzi

13.2k12136

$endgroup$

add a comment | 

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3

$begingroup$

Your are right, such plane is not unique. For example the planes $2x-y+3z=18$ and $5x+4y-z=-13$ pass through the point $(2,-5,3)$ and they are parallel to the given line.

More generally, through the given point, there is a unique line parallel to the given line, but then any plane through this second line is parallel to the given line.

share|cite|improve this answer

edited Mar 14 at 4:25

answered Mar 13 at 9:55

Robert ZRobert Z

101k1069142

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I'm thinking like this, but my teacher gave some explanation that I didn't really understand. She says that there is only one such plane
  $endgroup$
  – someone123123
  Mar 13 at 9:57
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Maybe the problem was "find a plane which passes through a given point and it is orthogonal to a given line.
  $endgroup$
  – Robert Z
  Mar 13 at 10:00
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Or perhaps the plane is supposed to include the intersection line of the two given planes (see updated question). Impossible to know without seeing the teacher’s solution.
  $endgroup$
  – amd
  Mar 14 at 0:24
  

  
  
  
  

add a comment | 

3

$begingroup$

Your are right, such plane is not unique. For example the planes $2x-y+3z=18$ and $5x+4y-z=-13$ pass through the point $(2,-5,3)$ and they are parallel to the given line.

More generally, through the given point, there is a unique line parallel to the given line, but then any plane through this second line is parallel to the given line.

share|cite|improve this answer

edited Mar 14 at 4:25

answered Mar 13 at 9:55

Robert ZRobert Z

101k1069142

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I'm thinking like this, but my teacher gave some explanation that I didn't really understand. She says that there is only one such plane
  $endgroup$
  – someone123123
  Mar 13 at 9:57
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Maybe the problem was "find a plane which passes through a given point and it is orthogonal to a given line.
  $endgroup$
  – Robert Z
  Mar 13 at 10:00
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Or perhaps the plane is supposed to include the intersection line of the two given planes (see updated question). Impossible to know without seeing the teacher’s solution.
  $endgroup$
  – amd
  Mar 14 at 0:24
  

  
  
  
  

add a comment | 

$begingroup$

Your are right, such plane is not unique. For example the planes $2x-y+3z=18$ and $5x+4y-z=-13$ pass through the point $(2,-5,3)$ and they are parallel to the given line.

More generally, through the given point, there is a unique line parallel to the given line, but then any plane through this second line is parallel to the given line.

share|cite|improve this answer

edited Mar 14 at 4:25

answered Mar 13 at 9:55

Robert ZRobert Z

101k1069142

$endgroup$

share|cite|improve this answer

edited Mar 14 at 4:25

answered Mar 13 at 9:55

Robert ZRobert Z

101k1069142

answered Mar 13 at 9:55

Robert ZRobert Z

101k1069142

answered Mar 13 at 9:55

Robert ZRobert Z

101k1069142

1

$begingroup$

You are right: there are infinitely mane planes passing through a point and parallel to a given line.

share|cite|improve this answer

answered Mar 13 at 9:55

AlessioDVAlessioDV

955114

$endgroup$

add a comment | 

1

$begingroup$

You are right: there are infinitely mane planes passing through a point and parallel to a given line.

share|cite|improve this answer

answered Mar 13 at 9:55

AlessioDVAlessioDV

955114

$endgroup$

add a comment | 

$begingroup$

You are right: there are infinitely mane planes passing through a point and parallel to a given line.

share|cite|improve this answer

answered Mar 13 at 9:55

AlessioDVAlessioDV

955114

$endgroup$

share|cite|improve this answer

answered Mar 13 at 9:55

AlessioDVAlessioDV

955114

answered Mar 13 at 9:55

AlessioDVAlessioDV

955114

answered Mar 13 at 9:55

AlessioDVAlessioDV

955114

1

$begingroup$

As another answer points out, the claim is false. Given a set of planes parallel to each other as well as to the given line, only one of those planes will pass through the given point.

share|cite|improve this answer

edited Mar 13 at 10:06

answered Mar 13 at 10:00

Oscar LanziOscar Lanzi

13.2k12136

$endgroup$

add a comment | 

1

$begingroup$

As another answer points out, the claim is false. Given a set of planes parallel to each other as well as to the given line, only one of those planes will pass through the given point.

share|cite|improve this answer

edited Mar 13 at 10:06

answered Mar 13 at 10:00

Oscar LanziOscar Lanzi

13.2k12136

$endgroup$

add a comment | 

$begingroup$

As another answer points out, the claim is false. Given a set of planes parallel to each other as well as to the given line, only one of those planes will pass through the given point.

share|cite|improve this answer

edited Mar 13 at 10:06

answered Mar 13 at 10:00

Oscar LanziOscar Lanzi

13.2k12136

$endgroup$

share|cite|improve this answer

edited Mar 13 at 10:06

answered Mar 13 at 10:00

Oscar LanziOscar Lanzi

13.2k12136

answered Mar 13 at 10:00

Oscar LanziOscar Lanzi

13.2k12136

answered Mar 13 at 10:00

Oscar LanziOscar Lanzi

13.2k12136