Plane Plane Intersection Question

[markraz]

Hi, so I have 2 planes
2x + 1y +3z = 4
3x + -1y +4z = 6

I want to find their intersection



I am using this formula: (A^t * A)^-1 * A^t * b

when solved the answer is:

x= 2.5
y= -.375
z= .5

my question is what does the answer tell me?

 

[Romsek]

You should have enough intuition by now to know that two planes cannot intersect at a point.

If we add the two equations we get

[MATH]5x + 7z = 10\\ z = -\dfrac 5 7 x + \dfrac{10}{7}[/MATH]
And this is the equation of the line of intersection.

What you did looks like some sort of least squares fit. Where did you get that formula?

 

[markraz]

You should have enough intuition by now to know that two planes cannot intersect at a point.

thanks, yeah that is why I am asking the question

What you did looks like some sort of least squares fit. Where did you get that formula?

it's called "the normal equation". I got it on stackoverflow .com

If we add the two equations we get

thanks but I need to use matricies. I am going to be calculating a few million intersections and I won't be able to do division since it is way too slow.
can I do this with matrix? fractions are no good for what I am going to be doing

thanks

 

[Steven G]

2x + 1y +3z = 4
3x + -1y +4z = 6

[math]\begin{equation} \begin{matrix} 2 & 1 & 3 & | & 4 \\ 3 & {-1} & 4 & | & 6 \\ \end{matrix} \end{equation}[/math]
You can add the two rows, if that is what you want.
You need to understand that whatever you can do with the equations you can do with matrices.

 

[Romsek]

This is about as simple as it gets.

http://www.math.uwaterloo.ca/~jmckinno/Math106/Week3/Lecture1s.pdf

 

[markraz]

This is about as simple as it gets.

http://www.math.uwaterloo.ca/~jmckinno/Math106/Week3/Lecture1s.pdf

thnaks that works great

 

[markraz]

2x + 1y +3z = 4
3x + -1y +4z = 6

[math]\begin{equation} \begin{matrix} 2 & 1 & 3 & | & 4 \\ 3 & {-1} & 4 & | & 6 \\ \end{matrix} \end{equation}[/math]
You can add the two rows, if that is what you want.
You need to understand that whatever you can do with the equations you can do with matrices.

thanks