The definition of the Plane $\pi$ says that given a point $A$ $\in \pi$ and a normal vector $\vec{n}$ a point $P \in \pi$ if and only if $\vec{AP} \cdot \vec{n} = 0$, from that we can get $\vec{n} \cdot P – \vec{n} \cdot A = 0$.
Now my doubt is, can we have Dot Product between a vector and a point? Or isn't that a point? What am I missing?
Best Answer
If you distinguish points and vectors (i.e. consider them elements of different sets) and don't introduce the product of a point with a vector (which would indeed be a strange thing to introduce in that case), it just means that you cannot apply the distributive law. That is, you cannot rewrite $\vec n\cdot (P-A)$ as $\vec n\cdot P - \vec n\cdot A$.
However the more common way to deal with it is to select an arbitrary point $O$, called origin, and then to identify all points with their "location vectors" $\vec P = P - O$ (obviously, $\vec O = O - O = \vec 0$). Now of course $\vec n\cdot \vec P$ is well defined, although in general not a meaningful quantity to calculate in isolation, as its value depends on the arbitrary choice of $O$. However $\vec n\cdot \vec P - \vec n\cdot\vec A$ does not depend on the choice of origin and therefore is a meaningful quantity to calculate.