A plane in 3D coordinate space is determined by a point and a vector that is perpendicular to the plane.
Let \(P_0 = (x_0, y_0, z_0) \) be the point given,
and \(n\) the orthogonal vector.
Also let \(P = (x, y, z)\) be any point in the plane, and \(r\) and \(r_0\) the position vectors of
points \(P\) and \(P_0\), respectively.
Now, if we let \(n = (a,b,c) \), then since \(\overrightarrow{P_0 P} \) is perpendicular to \(n\), we have
\[
\,\\
\begin{align}
\overrightarrow{P_0 P} \cdot n &= (P - P_0) \cdot n \\
&= (x - x_0, y - y_0, z - z_0) \cdot (a, b, c) \\
&= a(x - x_0) + b(y - y_0) + c(z - z_0) \\
&= ax + by + cz - (ax_0 + by_0 + cz_0) \\
\end{align}
\,\\
\]
This expression must be zero for all points \( P \) on the plane,
because the vector \( \overrightarrow{P_0P} \) is orthogonal to \( n \).
We define the constant term
\[
\,\\
d = -(ax_0 + by_0 + cz_0)
\,\\
\]
and thus obtain the plane equation in its scalar form:
\[
\,\\
ax + by + cz + d = 0
\,\\
\]
In vector form, the plane can be written as below if we let \(p = P - P_0\)
\[
\,\\
p \cdot n = 0
\,\\
\]
That is, for every point \( P \) on the plane, the vector \( P - P_0 \) is orthogonal to \( n \),
meaning the dot product between \( n \) and \( P - P_0 \) is \(0\).
Conversely, any point that satisfies this condition must lie on the plane defined by \( n \) and \( P_0 \).