# Thread: 2 dimensional plane in four dimensions

1. ## 2 dimensional plane in four dimensions

In R4 find the equation of the 2 dimensional plane that passes through the point (0,1,1,-2) and is normal to the vectors N = (2, -1, 0, 1) and M = (0, -1, 3, 0).

I know that you can define a plane as (P-v).N = 0 and (P-v).M=0. What do I have to do in this case?

2. Originally Posted by TTim
In R4 find the equation of the 2 dimensional plane that passes through the point (0,1,1,-2) and is normal to the vectors N = (2, -1, 0, 1) and M = (0, -1, 3, 0).

I know that you can define a plane as (P-v).N = 0 and (P-v).M=0. What do I have to do in this case?
choose any point $Q=(x,y,z,t)$ on the plane and let $P=(0,1,1,-2).$ then $\vec{PQ}=(x,y-1,z-1,t+2)$ and we want to have $\vec{PQ} \cdot M=\vec{PQ} \cdot N= 0,$ which

gives us: $2x-y+t+3=y-3z+2=0.$ thus $y=3z-2, \ t=3z-2x-5.$ hence the vector equation of the 2-dimensional plane that you're looking for is:

$(0,-2,0,-5)+(1,0,0,-2)x + (0,3,1,3)z,$ where $x,z$ are any real numbers.