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July 25, 2025, 11:35:06 am

Author Topic: Plane - Linear algebra  (Read 1632 times)  Share 

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QuantumJG

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Plane - Linear algebra
« on: September 21, 2009, 02:10:52 pm »
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Find the equation of a plane (in both cartesian and parametric form) that passes through the point (1,6,-4) and contains the line:

x = 1 + 2t, y = 2 - 3t, z = 3 - t, t element of R

My problem is how can you use the line to derive the equation of the plane?
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QuantumJG

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Re: Plane - Linear algebra
« Reply #1 on: September 21, 2009, 03:06:17 pm »
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Find the equation of a plane (in both cartesian and parametric form) that passes through the point (1,6,-4) and contains the line:

x = 1 + 2t, y = 2 - 3t, z = 3 - t, t element of R

My problem is how can you use the line to derive the equation of the plane?

Don't worry I got it!
2008: Finished VCE

2009 - 2011: Bachelor of Science (Mathematical Physics)

2012 - 2014: Master of Science (Applied Mathematics/Mathematical Physics)

2016 - 2018: Master of Engineering (Civil)

Semester 1:[/b] Engineering Mechanics, Fluid Mechanics, Engineering Risk Analysis, Sustainable Infrastructure Engineering

Semester 2:[/b] Earth Processes for Engineering, Engineering Materials, Structural Theory and Design, Systems Modelling and Design

Damo17

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Re: Plane - Linear algebra
« Reply #2 on: September 21, 2009, 03:07:40 pm »
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Find the equation of a plane (in both cartesian and parametric form) that passes through the point (1,6,-4) and contains the line:

x = 1 + 2t, y = 2 - 3t, z = 3 - t, t element of R

My problem is how can you use the line to derive the equation of the plane?


Well we know that the points and are apart of the plane. We now need another point in the plane so we can find the normal. So let , and we get the point .

So now let:
a be the vector from to and b be the vector from to .
We now get: and .

The normal:
(forgive me here, these are meant to represent matrices with the first 2 numbers of each in the first column, then the last two numbers the second column.)

i component
j component
k component

so


you now use to get:
Equation of plane:
« Last Edit: September 21, 2009, 03:19:52 pm by Damo17 »
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kamil9876

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Re: Plane - Linear algebra
« Reply #3 on: September 21, 2009, 03:12:17 pm »
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If you imagine it, the information about a line on it's own will not be sufficient information for the plane, but the information about another point(not on this line) narrows it down to one plane.

we know that the parametric form of the plane is as follows:

where

we can make (ie it's in the plane and line, you get it from t=0).

we can make since it's the direction vector of the line, so it's a direction vector of the plane too.

The final direction vector can be obtains from "connecting" a point on the line to the additional point given, (1,6,-4). This gives:



Hence the parametric equation of the line is:


Cartesian equation can be derived from here by finding a normal vector.
« Last Edit: September 21, 2009, 03:24:34 pm by kamil9876 »
Voltaire: "There is an astonishing imagination even in the science of mathematics ... We repeat, there is far more imagination in the head of Archimedes than in that of Homer."

QuantumJG

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Re: Plane - Linear algebra
« Reply #4 on: September 21, 2009, 03:23:22 pm »
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If you imagine it, the information about a line on it's own will not be sufficient information for the plane, but the information about another point(not on this line) narrows it down to one plane.

we know that the parametric form of the plane is as follows:

where

we can make (ie it's in the plane and line, you get it from t=0).

we can make since it's the direction vector of the line, so it's a direction vector of the plane too.

The final direction vector can be obtains from "connecting" a point on the line to the additional point given, (1,6,-4). This gives:



Hence the parametric equation of the line is:


Cartesian equation can be derived from here by finding a normal vector.

Yep after about an hour of thinking I realised that method was the way to go.

I got r = (1,2,3) + u(2,-3,-1) + s(0,4,-7)
2008: Finished VCE

2009 - 2011: Bachelor of Science (Mathematical Physics)

2012 - 2014: Master of Science (Applied Mathematics/Mathematical Physics)

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Semester 1:[/b] Engineering Mechanics, Fluid Mechanics, Engineering Risk Analysis, Sustainable Infrastructure Engineering

Semester 2:[/b] Earth Processes for Engineering, Engineering Materials, Structural Theory and Design, Systems Modelling and Design

kamil9876

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Re: Plane - Linear algebra
« Reply #5 on: September 21, 2009, 03:25:17 pm »
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yeah just realised a small mistake. 6-2=4 obsviously  :idiot2:
Voltaire: "There is an astonishing imagination even in the science of mathematics ... We repeat, there is far more imagination in the head of Archimedes than in that of Homer."