# Conditions for Homogeneity/Plane

## Theorem

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

## Proof

### 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 $P$.

Hence:

 $\ds \alpha_1 0 + \alpha_2 0 + \alpha_3 0$ $=$ $\ds \gamma$ $\ds \leadsto \ \$ $\ds \gamma$ $=$ $\ds 0$

$\Box$

### Necessary Condition

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

Then:

$0 = \alpha_1 0 + \alpha_2 0 + \alpha_3 0 \in P$

and so:

$\tuple {0, 0, 0}$ lies on $P$

Hence $P$ is homogeneous.

$\blacksquare$