Condition for Planes to be Parallel

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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:

$P' = \left\{{ \left({x_1, x_2, x_3}\right) \in \R^3 : \alpha_1 x_1 + \alpha_2 x_2 + \alpha_3 x_3 = \gamma' }\right\}$


Proof

Necessary Condition


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.

$\blacksquare$


Also see


Sources