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We are given that \( ACE \) is an equilateral triangle, and vectors \( \vec{u} \) and \( \vec{v} \) are defined as:

- \( \vec{AB} = \vec{u} \)
- \( \vec{AF} = \vec{v} \)

Let's break down the required vector expressions in terms of \( \vec{u} \) and \( \vec{v} \).

### Part a: \( \vec{AC} \)

Since \( B \) is the midpoint of \( AC \), we know that:

\[
\vec{AB} = \vec{u} \quad \text{and} \quad \vec{BC} = \vec{u}
\]

Thus, the vector \( \vec{AC} \) is:

\[
\vec{AC} = \vec{AB} + \vec{BC} = \vec{u} + \vec{u} = 2\vec{u}
\]

### Part b: \( \vec{AD} \)

\( D \) is the midpoint of \( CE \). From the diagram, \( \vec{AD} \) can be expressed using known vectors. First, we decompose \( \vec{AD} \) as:

\[
\vec{AD} = \vec{AF} + \vec{FD}
\]

Since \( FD \) is parallel and equal to \( AB \) (because of the symmetry of the equilateral triangle), and \( AB = \vec{u} \), we have:

\[
\vec{AD} = \vec{v} + \vec{u}
\]

### Part c: \( \vec{CE} \)

Since \( D \) is the midpoint of \( CE \) and triangles are equilateral, we know:

\[
\vec{CD} = \vec{DE}
\]

We can write:

\[
\vec{CE} = \vec{CD} + \vec{DE} = \text{(a vector symmetry identical to \( \vec{AC} \))} = 2\vec{v}
\]

### Part d: \( \vec{EB} \)

We can express \( \vec{EB} \) using the vectors as:

\[
\vec{EB} = \vec{EA} + \vec{AB}
\]

Since \( \vec{EA} = -\vec{v} \) (because it's in the opposite direction of \( \vec{AF} \)) and \( \vec{AB} = \vec{u} \), we have:

\[
\vec{EB} = -\vec{v} + \vec{u} = \vec{u} - \vec{v}
\]

---

### Part e: Give an example of a vector equal to \( 2\vec{v} \)

The vector \( \vec{CE} \) is already equal to \( 2\vec{v} \), so:

\[
\boxed{\vec{CE} = 2\vec{v}}
\]

### Part f: Give an example of a vector equal to \( \vec{u} - \vec{v} \)

The vector \( \vec{EB} \) is already equal to \( \vec{u} - \vec{v} \), so:

\[
\boxed{\vec{EB} = \vec{u} - \vec{v}}
\]

---

Do you need any more details or clarification on any of these steps?

Here are 5 additional related questions for further understanding:

1. How do you calculate vector magnitudes when given components in terms of \( \vec{u} \) and \( \vec{v} \)?
2. How would the equations change if the triangle were not equilateral?
3. What is the geometric interpretation of \( \vec{u} + \vec{v} \) and \( \vec{u} - \vec{v} \) in this context?
4. How can we verify that the triangles formed by midpoints (like \( BDF \)) are equilateral?
5. Can you find the area of triangle \( ACE \) using vector cross products?

**Tip**: When dealing with vectors in geometric shapes, identifying midpoints or symmetry often helps to simplify vector decomposition.

In the diagram, ACE is an equilateral triangle. Vertices B, D, and F are the midpoints of AC, CE, and EA, and AB = u and AF = v. Write the following in terms of u and v: AC, AD, CE, EB. Give an example of a vector that is equal to 2v and u - v.

Learn how to express various vectors in terms of u and v in an equilateral triangle. This problem covers vector addition, symmetry, and midpoint properties. Suitable for students in Grades 10-12.

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