# Thread: Finding the equation of a Plane

1. ## Finding the equation of a Plane

Hi, another vector question I'm afraid

I have the 4 lines:

$\displaystyle BC = \begin{pmatrix} -2 \\ -4 \\ 1 \end{pmatrix} + \lambda\begin{pmatrix} 3 \\ 6 \\ -3 \end{pmatrix}$, $\displaystyle AC = \begin{pmatrix} 1 \\ 2 \\ -2 \end{pmatrix} + \sigma\begin{pmatrix} -2 \\ 2 \\ -4 \end{pmatrix}$.

$\displaystyle l2 = \begin{pmatrix} 3 \\ 0 \\ 2 \end{pmatrix} + \beta\begin{pmatrix} -2 \\ 2 \\ -4 \end{pmatrix}$, $\displaystyle AD = \begin{pmatrix} 0 \\ 0 \\ -1 \end{pmatrix} + \alpha\begin{pmatrix} -3 \\ 0 \\ -3 \end{pmatrix}$.

I need to show that all 4 lines lie in a single plane and fine the Cartesian Equation of the plane.

The Cartesian equation of a plane is in the form $\displaystyle Ax+By+Cz=d$, so from this I can set up the following equations:

$\displaystyle -2A-4B+C=d$

$\displaystyle 3A+2C=d$

$\displaystyle A+2B-2C=d$

From these I can get all $\displaystyle A,B,C$ in terms of $\displaystyle d$.

$\displaystyle A=d$, $\displaystyle B=C=-d$

Putting these values back into the original Cartesian Equation I got:

$\displaystyle dx-dy-dz = d$, which can be simplified to $\displaystyle x-y-z=1$.

Is this correct so far or have I gone horribly wrong?

Thanks

2. Over 20 views and none of you have any ideas

Anyone got any comments at all?

3. You are correct so far.
With each line there is a point and a direction vector
Test each line with with respect to the plane $\displaystyle x-y-z=1$.
Does the point satisify that equation?
Is the direction vector perpendicular to $\displaystyle <1,-1,-1>?$

If the answer is yes in all cases you are done.

4. Originally Posted by Plato
You are correct so far.
With each line there is a point and a direction vector
Test each line with with respect to the plane $\displaystyle x-y-z=1$.
Does the point satisify that equation?
Is the direction vector perpendicular to $\displaystyle <1,-1,-1>?$

If the answer is yes in all cases you are done.
Yes and yes.

I was pretty sure that I'd gone about it in the right way but just wanted to make sure.