48. A perpendicular [OP] is drawn from the origin to a given plane, defining the distance from this plane to the origin. Let's introduce the following notations: $\vec{p}=\overrightarrow{O P},|\vec{p}|=p-$ the mentioned distance; $\vec{e}$-the unit vector of $\vec{p}$ (then $\vec{p}=p \vec{e}$ ); $\alpha, \beta, \gamma-$ the angles between
the unit vector $\vec{e}$ and the coordinate axes. Similarly to how the equation of a line in coordinates was derived on a plane, derive the equation of a plane in space:

a) in vector form: $\vec{r} \vec{e}=p$;

b) in coordinate form:

$$
x \cos \alpha+y \cos \beta+z \cos \gamma=p
$$

The last equation will be called the normal equation of the plane.
48. Let $\vec{r}$ be a variable radius vector. Since the vector $(\vec{r}-\vec{p})$ is perpendicular to the vector $\vec{p}$, we obtain the equation: $(\vec{r}-\vec{p}) \vec{p}=0 ; \quad \vec{r} \vec{e}=p . \quad$ Here $\quad \vec{r}=x \vec{i}+y \vec{j}+z \vec{k}, \quad \vec{p}=p \vec{e}$, $\vec{e}=\vec{i} \cos \alpha+\vec{j} \cos \beta+\vec{k} \cos \gamma$. Writing the relation $\vec{r} \vec{e}=\vec{p}$ in coordinates, we get: $x \cos \alpha+y \cos \beta+z \cos \gamma=p$ - the normal equation of the plane.