Problem

The points $A$, $B$ and $C$ have coordinates $(3,2,2)$, $(-1,1,3)$ and $(-2,4,2)$ respectively.

The plane $\Pi_1$ contains the points $A$, $B$ and $C$

(a) Determine a Cartesian equation of $\Pi_1$

Given that

• point $D$ has coordinates $(-1,1,-2)$

• line $l$ passes through $D$ and is perpendicular to $\Pi_1$

• plane $\Pi_2$ has equation $\mathbf r\cdot(14\mathbf i-\mathbf j-17\mathbf k)=-66$

• $l$ meets $\Pi_2$ at the point $E$

(b) show that $DE=p\sqrt{22}$ where $p$ is a rational number to be determined.

The point $F$ has coordinates $(4,3,q)$ where $q$ is a constant.

Given that $A$, $B$, $C$ and $F$ are the vertices of a tetrahedron of volume 12

(c) determine the possible values of $q$