Condition for Planes to be Parallel

Theorem
Let $P: \alpha_1 x_1 + \alpha_2 x_2 + \alpha_3 x_3 = \gamma$ be a plane in $\R^3$.

Then the plane $P'$ is parallel to $P$ iff there is a $\gamma' \in \R$ such that:
 * $P' = \left\{{ \left({x_1, x_2, x_3}\right) \in \R^3 : \alpha_1 x_1 + \alpha_2 x_2 + \alpha_3 x_3 = \gamma' }\right\}$

Sufficient Condition
Let $P' \ne P$ be a plane given by the equation:


 * $\alpha_1 x_1 + \alpha_2 x_2 + \alpha_3 x_3 = \gamma'$

Suppose we have a point $\vec x = \left({x_1, x_2, x_3}\right) \in P \cap P'$.

Then, as $\vec x \in P$, it also satisfies:


 * $\alpha_1 x_1 + \alpha_2 x_2 + \alpha_3 x_3 = \gamma$

It follows that $\gamma = \gamma'$, so $P = P'$.

This contradiction shows that $P \cap P' = \varnothing$, i.e., $P$ and $P'$ are parallel.

The remaining case is when $P' = P$. By definition, $P$ is parallel to itself.

The result follows.

Also See

 * Condition for Straight Lines being Parallel