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We are given the following:

- \(\vec{AB} = \vec{a}\), \(\vec{AC} = \vec{b}\)
- \(\vec{CD} = 2 \vec{BC}\)
- \(\vec{NC} \parallel \vec{AB}\)
- \(\vec{AN} = k \vec{AD}\)

From the problem, we also know:
- \(\vec{BC} = \vec{b} - \vec{a}\)
- \(\vec{AD} = 3\vec{b} - 2\vec{a}\)

We are tasked with finding the value of \(k\).

### Step-by-Step Solution:

1. **Use the given relation for \(\vec{CD}\):**
   Since \(\vec{CD} = 2\vec{BC}\), and we know that \(\vec{BC} = \vec{b} - \vec{a}\), we substitute:
   \[
   \vec{CD} = 2(\vec{b} - \vec{a}) = 2\vec{b} - 2\vec{a}
   \]
   
2. **Find the vector \(\vec{ND}\):**
   The point \(N\) lies on \(AD\), and we are given that \(\vec{AN} = k \vec{AD}\). This means:
   \[
   \vec{AN} = k(3\vec{b} - 2\vec{a})
   \]
   Hence, the vector \(\vec{ND}\) is:
   \[
   \vec{ND} = \vec{AD} - \vec{AN} = (3\vec{b} - 2\vec{a}) - k(3\vec{b} - 2\vec{a}) = (1 - k)(3\vec{b} - 2\vec{a})
   \]
   
3. **Use the condition that \(\vec{NC} \parallel \vec{AB}\):**
   Since \(\vec{NC} \parallel \vec{AB}\), we know that \(\vec{NC} = \lambda \vec{AB}\) for some scalar \(\lambda\), where \(\vec{AB} = \vec{a}\). So,
   \[
   \vec{NC} = \lambda \vec{a}
   \]
   
4. **Find \(\vec{NC}\):**
   From the diagram, \(\vec{NC} = \vec{AC} - \vec{AN}\), so:
   \[
   \vec{NC} = \vec{b} - k(3\vec{b} - 2\vec{a}) = \vec{b} - k(3\vec{b}) + k(2\vec{a}) = (1 - 3k)\vec{b} + 2k\vec{a}
   \]
   
5. **Equate \(\vec{NC}\) to \(\lambda \vec{a}\):**
   From step 3, \(\vec{NC} = \lambda \vec{a}\). Therefore, we equate:
   \[
   (1 - 3k)\vec{b} + 2k\vec{a} = \lambda \vec{a}
   \]
   For the vector equation to hold, the coefficients of \(\vec{a}\) and \(\vec{b}\) must match separately. So, the coefficient of \(\vec{b}\) gives:
   \[
   1 - 3k = 0 \quad \Rightarrow \quad k = \frac{1}{3}
   \]
   
Thus, the value of \(k\) is \(\boxed{\frac{1}{3}}\).

Would you like a further breakdown of any steps or have questions?

### Additional questions to explore:
1. How do you calculate the position vector of point \(N\) given \(k\)?
2. Why is the condition \(\vec{NC} \parallel \vec{AB}\) crucial for determining \(k\)?
3. What would change if \(\vec{CD} \neq 2 \vec{BC}\)?
4. How is the vector \(\vec{AD}\) calculated from \(\vec{AB}\) and \(\vec{AC}\)?
5. How do we check if two vectors are parallel using their components?

### Tip:
Always break down vector equations into their components to handle parallel or perpendicular conditions easily.

On the diagram below, \(\vec{AB} = \vec{a}\) and \(\vec{AC} = \vec{b}\) and \(\vec{CD} = 2\vec{BC}\). N is the point on AD such that \(\vec{AN} = k\vec{AD}\) and \(\vec{NC}\) is parallel to \(\vec{AB}\). Find the value of k.

Learn how to solve a vector geometry problem where you determine the value of k given vector relationships and conditions of parallelism. This problem involves key concepts such as vector addition, parallel vectors, and algebraic manipulation. Suitable for students in Grades 11-12.

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