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What does $a$ stand for in the following formula for the distance of a point to a plane?

$$h = |PF| = \frac{|d - p \cdot n|}{|n|} = \frac{|(a-p)\cdot n|}{|n|} .$$

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up vote 2 down vote accepted

The formula describes the distance of a point $P$ from a plane $\pi$. From my understanding, the symbols in the formula have the following meaning.

  • $p$ is the position vector of $P$.

  • $F$ is the projection of $P$ on the plane $\pi$. So $h = |PF|$ is the distance of $P$ from $\pi$.

  • The plane $\pi$ itself is given by the equation $(x-a) \cdot n = 0$, where $n$ is the normal vector to $\pi$, and $a$ is an arbitrary point on $\pi$. (One point to keep in mind is that $a$ is not uniquely determined by the equation of the plane.)

  • The quantity $d$ is defined to be $a \cdot n$. The geometric meaning of $d$ is that $|d|$ is the distance of the origin from $\pi$.

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