Conditions for Homogeneity/Plane

Theorem
The plane $P = \alpha_1 x_1 + \alpha_2 x_2 + \alpha_3 x_3 = \gamma$ is homogeneous $\gamma = 0$.

Sufficient Condition
Let the plane $P = \alpha_1 x_1 + \alpha_2 x_2 + \alpha_3 x_3 = \gamma$ be homogeneous.

Then the origin $\tuple {0, 0, 0}$ lies on the plane $P$.

That is, $\alpha_1 0 + \alpha_2 0 + \alpha_3 0= \gamma \implies \gamma = 0$.

Necessary Condition
Let the equation of $P$ be $P = \alpha_1 x_1 + \alpha_2 x_2 + \alpha_3 x_3 = 0$.

Then $0 = \alpha_1 0 + \alpha_2 0 + \alpha_3 0 \in P$ and so $\tuple {0, 0, 0}$ lies on the plane $P$.

Hence $P$ is homogeneous.