# Line of Intersection between two planes

• Jun 23rd 2011, 01:44 PM
IanCarney
Line of Intersection between two planes
How would I determine the line of intersection between plane 1: $4x + 2y + 6z -14 = 0$ and plane 2: $x - y + z - 5 = 0?$?

I don't remember exactly how to do it, and trying to find an explanation online just confused me even more :(

I was thinking somewhere along the lines of using the cross products of the normals, but I'm not sure what to do after that.
• Jun 23rd 2011, 01:48 PM
Plato
Re: Line of Intersection between two planes
Quote:

Originally Posted by IanCarney
How would I determine the line of intersection between plane 1: $4x + 2y + 6z -14 = 0$ and plane 2: $x - y + z - 5 = 0?$?
I was thinking somewhere along the lines of using the cross products of the normals

That cross product is now the direction vector for the line of intersection.

Now we need to find some point that is on both planes.
• Jun 23rd 2011, 01:50 PM
IanCarney
Re: Line of Intersection between two planes
Quote:

Originally Posted by Plato
That cross product is now the direction vector for the line of intersection.

Now we need to find some point that is on both planes.

Cross product would be (8, 2, -6), right?

For the point, are you supposed to let a variable equal zero and then find the other two variables?
• Jun 23rd 2011, 01:56 PM
Plato
Re: Line of Intersection between two planes
So far, so good.
• Jun 23rd 2011, 01:58 PM
IanCarney
Re: Line of Intersection between two planes
Okay so assuming I let $z=0$

$4x+2y=14$ and $x-y=5$

$y=-1$ and $x=-4$

Would the equation of the line be $x=8t-4$, $y=2t-1$, and $z=-6t$?
• Jun 23rd 2011, 02:35 PM
IanCarney
Re: Line of Intersection between two planes
Ugh, according to the book the answer is $x=8+4t$, $y=t$, $z=-3-3t$

Not sure where I went wrong :s
• Jun 23rd 2011, 02:39 PM
Plato
Re: Line of Intersection between two planes
Quote:

Originally Posted by IanCarney
Ugh, according to the book the answer is $x=8+4t$, $y=t$, $z=-3-3t$

Those are different ways to write the same line.
Can you prove that?

EDIT: there may be a sign error in your point. Check the x value.
• Jun 23rd 2011, 02:48 PM
IanCarney
Re: Line of Intersection between two planes
Quote:

Originally Posted by Plato
Those are different ways to write the same line.
Can you prove that?

EDIT: there may be a sign error in your point. Check the x value.

Which x value are you talking about?

Edit: Oh right, it's supposed to be $+4$. Thanks for your help!
• Jun 23rd 2011, 06:15 PM
Soroban
Re: Line of Intersection between two planes
Hello, IanCarney!

I divided the first equation by 2 . . . Don't know why "they" didn't.

Here is a rather primitive method.

Quote:

Determine the line of intersection of planes: . $\begin{array}{cccc} 2x + y + 3z &=& 7 & [1] \\ x - y + z &=& 5 & [2]\end{array}$

Add [1] and [2]: . $3x + 4z \:=\:12 \quad\Rightarrow\quad x \:=\:4 - \tfrac{4}{3}z$

Substitute into [1]: . $2(4 - \tfrac{4}{3}z) + y + 3z \:=\:7 \quad\Rightarrow\quad y \:=\:\text{-}1 - \tfrac{1}{3}x$

We have: . $\begin{Bmatrix}x &=& 4 - \frac{4}{3}z \\ \\[-4mm]y &=& \text{-}1 - \frac{1}{3}z \\ \\[-4mm] z &=& z \end{Bmatrix}$

On the right, replace $z$ with the parameter $t.$

. . . . . . . . . $\begin{Bmatrix}x &=& 4 - \frac{4}{3}t \\ \\[-4mm] y &=& \text{-}1 - \frac{1}{3}t \\ \\[-4mm] z &=& t \end{Bmatrix}$

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

We can check our results.

. . . . . . . . . . . $2x + y + 3z \;=\;7$

. . $2(4-\tfrac{4}{3}) + (\text{-}1 - \tfrac{1}{3}z) + 3t \:=\:7$

. . . . . $8 - \tfrac{8}{3}y -1 - \tfrac{1}{3}t + 3t \;=\;7$

. - - - - . . . . . . . . . . . . . $7 \;=\;7 \;\;\;check!$

. . . . . . . . . . . . $x - y + z \;=\;5$

. . $(4-\tfrac{4}{3}t) - (\text{-}1-\tfrac{1}{3}t) + t \;=\;5$

. . . . . $4 - \tfrac{4}{3}t + 1 + \tfrac{1}{3}t + t \;=\;5$

. . . . . . . . . . . . . . . . . $5 \;=\;5 \;\;\;check!$