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 $\mathbf x = \left({x_1, x_2, x_3}\right) \in P \cap P\,'$.

Then, as $\mathbf x \in P$, 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 in Plane to be Parallel