How to prove mathematically that two planes parallel to a third plane are parallel

Question

Without relying on geometrical intuition and purely using vector calculus, how do we show that two planes parallel to a third plane are parallel?

I assume three dimensional space.

Answer 1

You can in general assume an $n$-dimensional space, call it $A$. Let $B=\{X;\vec{u_1},\dots{},\vec{u_k}\}$, $C=\{Y;\vec{v_1},\dots{},\vec{v_l}\}$ are any two subspaces of $A$. Spaces $A,B$ are said to be parallel, if $\langle{\vec{u_1},\dots{},\vec{u_k}\rangle}\subseteq\langle{\vec{v_1},\dots{},\vec{v_l}\rangle}$ or $\langle{\vec{v_1},\dots{},\vec{v_k}\rangle}\subseteq\langle{\vec{u_1},\dots{},\vec{u_l}\rangle}$. (At least that's what I was taught in several courses)

Now, if $P$ denotes all subspaces of $A$ of a given dimension $m\in\mathbb{N}$, it is evident, that the relation "being parallel" is equivalency on $P$, so the answer to your question now follows from it's transitivity.

Note that similar statement does not hold for a set $Q$ of all subspaces of $A$. Indeed, you can consider any plane (two-dimensional space). Then every line that lies in this plane is parallel to that line (using the definition above), but obviously not every two lines are parallel.

Answer 2

Embedded in $ \mathbb R ^3 $ if two planes P&Q are parallel, then distance along normal for all points is of constant length pq. ( We can drop perpendicular from P to Q or vice-versa).

Similarly Q&R, qr; P&R, pr ; the three planes are parallel to each other.

Answer 3

I don't know about vector calculus, but here's a good way of doing it:

Definition 0. Let $V$ denote a vector space over the reals. Then $A \subseteq V$ is a plane iff there exists a two-dimensional linear subspace $X$ of $V$, together with an element $v \in V$, such that $A = X+v$.

Definition 1. Let $V$ denote a vector space over the reals. Consider planes $A$ and $B$ of $V$. Then $A$ is parallel to $B$ iff there exists a vector $v$ such that $A+v=B$.

With these definitions, its pretty easy to prove that being parallel is an equivalence relation. Let me know if you have any trouble.