# Why are sheafs of planes linear combinations of two other planes

In most books, it is simply stated that the sheaf of planes can be described as a linear combination of two planes, but I cannot alight upon a simple explanation as to why.

I can see that a symmetric line equation in 3-D can be expressed as three symmetric 2-D line equations, that when plotted in 3-D describe three planes, and I can see that when you 'tweak' the multiples of these three symmetric equations to create a combined linear plane equation, the combined expression indeed rotates about the line common to all three planes. (Though one only needs two of these 2-D line equations when plotted in 3-D to produce the same effect.)

Any suggestions as to a simple explanation of why, at, say, the level of transition from school to university ?

• Could you please say a bit more about the context where your question arises? Particularly, the term "sheaf" has a technical meaning in mathematics that does not sound like the intended sense, and at the school-to-university level a "plane" is a subset of a Cartesian space, while "linear combination" at that level usually refers to vectors in a vector space. May 16, 2021 at 12:05
• Thanks. This is a standard exercise in the UK Cambridge (CAIE) A level second year Pure Maths syllabus 9709. In order to get the set of planes that share one line of intersection, the solution is to solve for a combination of the two plane equations. But nowhere can I find why one plane is actually a linear combination of another two. May 18, 2021 at 5:57

Here's one possible proof. Everything is in Euclidean three-space, equipped with a Cartesian coordinate system $$(x, y, z)$$.

Let $$P_{0} = (x_{0}, y_{0}, z_{0})$$ be a point, and $$\ell$$ a line through $$P_{0}$$. Suppose $$n_{1} = (a_{1}, b_{1}, c_{1})$$ and $$n_{2} = (a_{2}, b_{2}, c_{2})$$ are non-proportional vectors orthogonal to $$\ell$$. The equations \begin{align*} f_{1}(x, y, z) &:= a_{1}(x - x_{0}) + b_{1}(y - y_{0}) + c_{1}(z - z_{0}) = 0, \\ f_{2}(x, y, z) &:= a_{2}(x - x_{0}) + b_{2}(y - y_{0}) + c_{2}(z - z_{0}) = 0, \end{align*} define planes containing $$\ell$$. If $$t_{1}$$ and $$t_{2}$$ are real numbers, then $$t_{1}f_{1}(x, y, z) + tf_{2}(x, y, z) = 0$$ is the equation of the plane containing $$\ell$$ and with normal vector $$t_{1}n_{1} + t_{2}n_{2}$$. Since $$n_{1}$$ and $$n_{2}$$ are non-proportional, every non-zero vector orthogonal to $$\ell$$ may be written as a linear combination of $$n_{1}$$ and $$n_{2}$$. Since every plane containing $$\ell$$ is uniquely specified by a vector orthogonal to $$\ell$$, every plane containing $$\ell$$ is a linear combination of the given equations.