# Plane by plane-plane intersection and point

• Task: Find the equation of the plane containing the intersection line of E1 and E2 and the point P

$$E_1:\,\, 2x + y + 5z = 31$$

$$E_2:\,\, -4x + 5y + 4z = 50\,\,\,,\,\,P( -5 \,,\, 2 \,,\, 3 )$$

• Question: Is my approach for the following problem valid? If yes, why doesn't it yield the right result?

1.: Combine the plane equations $E_1$ and $E_2$, canceling out a variable

--> $$(2 * E_1) + E_2 \longrightarrow 7y + 14z= 112 \Longrightarrow y + 2z = 16$$

2.: Use the new 2D line equation above to find the y and z values of two points on the intersection line, using one of the original plane equations to find the corresponding x value

$$P_1( x \,,\, 0 \,,\, 8 ) \longrightarrow E_1: 2x + 0 + 40 = 31 \Longrightarrow A( -4.5 \,,\, 0 \,,\, 8 )\Longrightarrow$$ $$P_2( x \,,\, 2 \,,\, 7 ) \longrightarrow E_1:\,\, 2x + 2 + 35 = 31 \Longrightarrow B( -3 \,,\, 2 \,,\, 7 )$$

3.: Get two vectors containing the new plane: A to B and A to P

$AB = B - A \longrightarrow AB( 1.5 | 2 | -1 )$

$AP = P - A \longrightarrow AP( 9.5 | 1 | -10 )$

4.: Use the cross product of $AB$ and $AP$ to get the normal vector of the searched plane

My result: $n(19 | -5.5 | 17.5)$

Solution for the plane equation as given by teacher: $-2x + 2y + z = 17$

• Where did I go wrong?
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I found that P is not in the plane given in your teacher's solution. –  Siming Tu Apr 21 '12 at 14:28
@Siming: Damnit, I entered a wrong point P - fixed it now –  Ralf Apr 21 '12 at 14:35
Your work was then based off the incorrect $P$. Recalculate $P-A$ and the cross product. It will lead to the teacher's answer. –  David Mitra Apr 21 '12 at 14:37
I'm a little worried about Step 1. $y+2z=16$ is not the equation of a line, it's the equation of a plane. –  Gerry Myerson Aug 30 '12 at 13:02

As Gerry Myerson noted, the error occurs already at Step 1. The equation $y+2z=16$ describes a plane that contains the line of intersection. When you use it to find $P_1$ and $P_2$, you get two points in this plane, but not necessarily on the line. The logic of steps 3 and 4 is correct, but they are based on incorrect points, so the result is wrong.