3 Plane sheaf中直线的几何意义、三维空间表示及矩阵方程含义的技术问询（A level further maths）

Hey folks, I’m working through my A Level Further Maths and I’ve hit a wall with this problem about three planes forming a sheaf (they all intersect along a single straight line). I can’t find a straight answer anywhere, so let me lay out exactly what’s confusing me:

First, here’s the example system of planes I’m looking at:

(1) 3x - y - 6z = 1
(2) x + 3y + 3z = 2
(3) -3x - y + 3z = -1

I did some elimination to simplify things:

• Adding equation (1) to twice equation (2) gives 5x + 5y = 5 (let’s call this equation (4))
• Subtracting equation (3) from equation (2) gives 4x + 4y = 4 (equation (5))

I can see equations (4) and (5) are just scalar multiples of each other, so the system is consistent—all three planes meet along a straight line. I initially thought since there are 2 variables involved, the line is 2-dimensional, but wait, no—lines are 1-dimensional, right? Planes are 2D. That part I might have mixed up, but my main questions are:

• How is this line represented in 3D space? I know it’s a 1D object, but it exists within a 3D coordinate system—what’s the proper way to write its equation(s) to show its position and direction in 3D?
• The problem states each plane is represented by a matrix (x,y,z)—but if this matrix isn’t the normal vector of the plane (which I usually associate with plane equations), what does it actually signify?

I’ve also checked my textbook sections on this topic, but they aren’t clearing things up for me.

备注：内容来源于stack exchange，提问作者Kevin Zhai