Let's break down the problem and approach it step by step:

Problem Breakdown:

1. Triangle ABC:
   • $D$ is the midpoint of $AC$.
   • $E$ is a point on $BC$, and $BE:EC = 2:1$.
   • $F$ is a point on $BD$, with $BF:FD = 1:r$.
   • $EF$ is produced to meet $AB$ at $G$, with $EF:FG = 3:4$.
   • The position vectors are given as:
     • $\overrightarrow{OA} = 3i + 2j + k$
     • $\overrightarrow{OB} = 12i + 8j - 2k$
     • $\overrightarrow{OC} = 6i + j + k$
   • $O$ is the origin.

Part (a):

We need to express vectors $\overrightarrow{AF}$ and $\overrightarrow{AG}$ in terms of $r$.

Part (b):

1. (i) Find the dot product $\overrightarrow{AD} \cdot \overrightarrow{DB}$.
2. (ii) Concyclic question: Determine if points $A$, $D$, $F$, and $G$ are concyclic.

I'll now start with Part (a), expressing $\overrightarrow{AF}$ and $\overrightarrow{AG}$ in terms of $r$. First, I will calculate the coordinates of key points such as $D$, $E$, $F$, and $G$.

Let me compute that for you.The coordinates of the key points calculated so far are:

• $D$ (midpoint of $AC$):
  $\overrightarrow{OD} = \left( \frac{9}{2}, \frac{3}{2}, 1 \right)$

• $E$ (dividing $BC$ in the ratio $2:1$):
  $\overrightarrow{OE} \approx \left( 10, \frac{17}{3}, -1 \right)$

I will now proceed with the expression for $\overrightarrow{OF}$, which divides $BD$ in the ratio $1:r$, to find $F$. Then, I will proceed to express $\overrightarrow{AF}$ and $\overrightarrow{AG}$ in terms of $r$.The expression for the position vector of $F$ in terms of $r$ is:

$\overrightarrow{OF} = \frac{12 + 9r}{2(1 + r)} \hat{i} + \frac{8 + 3r}{2(1 + r)} \hat{j} + \frac{-2 + r}{1 + r} \hat{k}$

Next, I will proceed to find the position of $G$, which is where $EF$ (extended) meets $AB$, and express $\overrightarrow{AF}$ and $\overrightarrow{AG}$ in terms of $r$.It seems the computations for vectors $\overrightarrow{AF}$ and $\overrightarrow{AG}$ encountered an issue. Let me attempt a manual check on this. Would you like me to manually break down the steps for vector $\overrightarrow{AG}$ or perhaps proceed with part (b) for now?

Would you like to explore the following questions related to this topic?

1. How do we find the parametric equation of lines in space?
2. How do we prove four points are concyclic in geometry?
3. How can vector cross-products help in proving concyclicity?
4. How does the ratio theorem apply in dividing line segments?
5. How do dot products relate to angles between vectors?

Tip: Always visualize geometric problems with diagrams for clarity in vector-based problems.