The planes 3x+2y+z=6 and x+2y+5z=1- intersect along the line (x+2)/2 = (y-6)/(-7/2)= z.
A third plane passes through the origin and is perpendicular to the intersection of the first two planes, at what point do the three planes intersect?
Intersection in parametric eq:
x = -2 + 2t
y = 6 - 7/2t
z = t
The Attempt at a Solution
At first glance I knew that a normal vector will be perpendicular to the plane, so I just took the vector from the intersection <2, -7/2, 1> and used that as the normal vector for the third plane. I knew it passes through origin so i used point (0,0,0)
2x - 7/2y + z = D
I input (0,0,0) for (x,y,z) and get D= 0
Thus the equation for the third plane is
2x - 7/2y + z = 0
At this point I plot everything in maple and the intersection that was given to me doesnt even look like it's the true intersection, none the less my third plane doesn't look like its perpendicular to my intersection either.
We have never done 3 planes intersecting but I assume it similar to 2 planes, where I set a variable equal to zero such as x=0 and then solve the system of equations. Once the other 2 are found, we can then just find the third and that will be the point in the intersection. I am not sure how to find the vector