The equation of a plane $\Pi$ in $\mathbb{R}^{3}$ is

$$\mathbf{x} \cdot \mathbf{n}=d$$

where $d$ is a constant scalar and $\mathbf{n}$ is a unit vector normal to $\Pi$. What is the distance of the plane from the origin $O$ ?

A sphere $S$ with centre $\mathbf{p}$ and radius $r$ satisfies the equation

$$|\mathbf{x}-\mathbf{p}|^{2}=r^{2}$$

Show that the intersection of $\Pi$ and $S$ contains exactly one point if $|\mathbf{p} \cdot \mathbf{n}-d|=r$.

The tetrahedron $O A B C$ is defined by the vectors $\mathbf{a}=\overrightarrow{O A}, \mathbf{b}=\overrightarrow{O B}$, and $\mathbf{c}=\overrightarrow{O C}$with $\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c})>0$. What does the condition $\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c})>0$ imply about the set of vectors $\{\mathbf{a}, \mathbf{b}, \mathbf{c}\}$ ? A sphere $T_{r}$ with radius $r>0$ lies inside the tetrahedron and intersects each of the three faces $O A B, O B C$, and $O C A$ in exactly one point. Show that the centre $P$ of $T_{r}$ satisfies

$$\overrightarrow{O P}=r \frac{|\mathbf{b} \times \mathbf{c}| \mathbf{a}+|\mathbf{c} \times \mathbf{a}| \mathbf{b}+|\mathbf{a} \times \mathbf{b}| \mathbf{c}}{\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c})}$$

Given that the vector $\mathbf{a} \times \mathbf{b}+\mathbf{b} \times \mathbf{c}+\mathbf{c} \times \mathbf{a}$ is orthogonal to the plane $\Psi$ of the face $A B C$, obtain an equation for $\Psi$. What is the distance of $\Psi$ from the origin?