Vector Equation of Plane
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Theorem
Let $P$ be a plane which passes through a point $C$ whose position vector relative to the origin $O$ is $\mathbf c$.
Let $\mathbf p$ be the vector perpendicular to $P$ from $O$.
Let $\mathbf r$ be the position vector of an arbitrary point on $P$.
Then $P$ can be represented by the equation:
- $\mathbf p \cdot \paren {\mathbf r - \mathbf c} = 0$
where $\cdot$ denotes dot product.
Proof
It is seen that $\mathbf r - \mathbf c$ lies entirely within the plane $P$.
As $P$ is perpendicular to $\mathbf p$, it follows that $\mathbf r - \mathbf c$ is perpendicular to $\mathbf p$.
Hence by Dot Product of Perpendicular Vectors:
- $\mathbf p \cdot \paren {\mathbf r - \mathbf c} = 0$
$\blacksquare$
Sources
- 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text {II}$: The Products of Vectors: $2$. The Scalar Product