Distance From Point To Plane

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The perpendicular distance of point (x₁, y₁, z₁) from plane Ax + By + Cz + D = 0 is

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1. **Definition**: The length of the perpendicular from point $P(x_1, y_1, z_1)$ to the plane $Ax + By + Cz + D = 0$ is the shortest distance.
2. **Vector Formula**: If the plane is $\vec{r} \cdot \vec{n} + D = 0$ and the point is $\vec{a}$, then distance $d = \frac{|\vec{a} \cdot \vec{n} + D|}{[|\vec{n}|}$.
3. **Cartesian Expansion**: Substituting $\vec{a} = (x_1, y_1, z_1)$ and $\vec{n} = (A, B, C)$:
   $d = \frac{|Ax_1 + By_1 + Cz_1 + D|}{\sqrt{A^2 + B^2 + C^2}}$.

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Distance From Point To Plane

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