Equation of Plane/General Equation

Theorem

A plane $P$ is the set of all $\tuple {x, y, z} \in \R^3$, where:

$\alpha_1 x + \alpha_2 y + \alpha_3 z = \gamma$

where $\alpha_1, \alpha_2, \alpha_3, \gamma \in \R$ are given, and not all of $\alpha_1, \alpha_2, \alpha_3$ are zero.

Proof

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Also presented as

The general form of the equation of the plane can also be presented in the form:

$A x + B y + C z + D = 0$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text V$: Vector Spaces: $\S 28$: Linear Transformations
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): plane (in Cartesian coordinates)