以原点 O 为基准，点 N、P 和 Q 的位置向量分别为 $\mathbf{n}$、$\mathbf{p}$ 和 $\mathbf{q}$，且平面 $\Pi$ 的方程为 $\mathbf{r}\cdot\mathbf{n}=0$。已知 $\mathbf{p}\cdot\mathbf{n}=\mathbf{q}\cdot\mathbf{n}\ne 0$。

1. Express the direction vector of PQ and compute its scalar product with $\mathbf{n}$.

$\overrightarrow{PQ} \cdot \mathbf{n} = (\mathbf{q} - \mathbf{p}) \cdot \mathbf{n}$
$= \mathbf{q}\cdot\mathbf{n} - \mathbf{p}\cdot\mathbf{n}$

2. Show that the scalar product is zero and conclude perpendicularity.

Since it is given that $\mathbf{p}\cdot\mathbf{n} = \mathbf{q}\cdot\mathbf{n}$,
$\overrightarrow{PQ} \cdot \mathbf{n} = \mathbf{q}\cdot\mathbf{n} - \mathbf{q}\cdot\mathbf{n} = 0$
Since the scalar product is $0$, $\overrightarrow{PQ}$ is perpendicular to $\mathbf{n}$ (shown).

3. Deduce the geometrical relationship between line PQ and plane $\Pi$.

The normal vector to plane $\Pi$ is $\mathbf{n}$.
Because the direction vector of line PQ ($\overrightarrow{PQ}$) is perpendicular to the normal of $\Pi$, line PQ is parallel to plane $\Pi$.
For P to lie on $\Pi$, we must have $\mathbf{p}\cdot\mathbf{n} = 0$. However, we are given $\mathbf{p}\cdot\mathbf{n} \ne 0$, so P does not lie on $\Pi$.
Hence, the line PQ is strictly parallel to plane $\Pi$ and does not lie on it.

1. Identify two non-parallel direction vectors parallel to the required plane.

The new plane is perpendicular to $\Pi$, so the normal vector of $\Pi$, which is $\mathbf{n}$, is a direction vector parallel to the new plane.

Since the plane contains points $P$ and $Q$, the vector $\overrightarrow{PQ} = \mathbf{q} - \mathbf{p}$ is another direction vector parallel to the plane.

(From part (i), $\mathbf{n}$ and $\mathbf{q} - \mathbf{p}$ are perpendicular, so they are non-parallel and form a valid pair of direction vectors for a plane).

2. Form the parametric vector equation of the plane.

Using the position vector of point $P$ (which is $\mathbf{p}$) as a known point on the plane, the vector equation of the plane is given by:

$\mathbf{r} = \mathbf{p} + \lambda(\mathbf{q} - \mathbf{p}) + \mu\mathbf{n}$
where $\lambda, \mu \in \mathbb{R}$.