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Q.
Distance of a point with position vector a from a plane $r \cdot \hat{ n }=d$ is
Three Dimensional Geometry
Solution:
Consider a point $P$ with position vector a and a plane $\pi_1$ whose equation is $r \cdot \hat{n}=d$.
Consider a plane $\pi_2$ through $P$ parallel to the plane $\pi_1$. The unit vector normal to $\pi_2$ is $\hat{n}$. Hence, its equation is $(r-a) \cdot \hat{n}=0$ i.e. $r \cdot \hat{n}=a \cdot \hat{n}$
Thus, the distance ON' of this plane from the origin is $|a \cdot \hat{n}|$. Therefore, the distance $P Q$ from the plane $\pi_1$ is [Fig. (a)] i.e., $O N-O N=|d-a \cdot \hat{n}|$
which is the length of the perpendicular from a point to the given plane.
We may establish the similar results for Fig. (b).
