Thread: Help with planes/lines of intersection

1. Help with planes/lines of intersection

Can someone help with the following problem:

Find an equation for the plane that passes through the point
P(1,2,-2) and contains the line of intersection of the planes
3x - y - z = 1 and x + y - z = 2. Show your work.

2. Originally Posted by messianic
Can someone help with the following problem:

Find an equation for the plane that passes through the point
P(1,2,-2) and contains the line of intersection of the planes
3x - y - z = 1 and x + y - z = 2. Show your work.

First find the intersection line of the two given planes: adding both equations you get $\displaystyle x = \frac{1}{2}\,z+\frac{3}{4}$, and substituting this in equation 2 you get $\displaystyle y=\frac{1}{2}\,z+\frac{5}{4}$, so the line, in parametric form, is $\displaystyle \left\{\left(\frac{1}{2}\,t+\frac{3}{4}\,,\,\frac{ 1}{2}\,t+\frac{5}{4}\,,\,t\right)\,,\,t\in\mathbb{ R}\right\}$ $\displaystyle =\left(\frac{3}{4}\,,\,\frac{5}{4}\,,\,0\right)+t\ left(\frac{1}{2}\,,\,\frac{1}{2}\,,\,1\right)\,,\, t\in\mathbb{R}$.

Now choose two different vectors in this line, say $\displaystyle A=\left(\frac{3}{4}\,,\,\frac{5}{4}\,,\,0\right)\, ,\,with\,\,\,t=0$ , and $\displaystyle B=\left(\frac{1}{4}\,,\,\frac{3}{4}\,,\,-2\right)\,,\,with\,\,\,t=-1$, and then the plane you're looking for is $\displaystyle P + r\overrightarrow{PA}+s\overrightarrow{PB}\,,\,r\,, \,s\in\mathbb{R}$.

And this question belongs in pre-algebra-analytic geometry, not in calculus.

Tonio