The vectors `vec(AB)=vec(c), vec(BC)=vec(a), vec(CA)=vec(b)`, are the sides of a triangles ABC. Which of the following vectors represent (s) the median `vec(AD)`?
1.` (1)/(2) vec(a)+vec(c)`
2. `-(1)/(2)vec(b)+(1)/(2)vec(c)`
3. `vec(1)/(2) vec(a)+vec(b)`
Select the correct answer using the code given below

Options

A. `1 and 2`
B. `1 and 3`
C. `1 only`
D. `2 only`

Explanation

To find the vector representing the median \( \vec{AD} \) in triangle \( ABC \) with given vectors \( \vec{AB} = \vec{c} \), \( \vec{BC} = \vec{a} \), and \( \vec{CA} = \vec{b} \), we can follow these steps:

### Step 1: Understand the Position of Points
In triangle \( ABC \):
- Point \( A \) is at the top.
- Point \( B \) is at the bottom left.
- Point \( C \) is at the bottom right.

### Step 2: Identify the Midpoint
Since \( D \) is the midpoint of \( BC \), we can express the position vector of \( D \) as:
\[ \vec{D} = \frac{1}{2} (\vec{B} + \vec{C}) \]
Using the vector representation, we can express \( \vec{B} \) and \( \vec{C} \) in terms of \( \vec{A} \):
- \( \vec{B} = \vec{A} + \vec{c} \)
- \( \vec{C} = \vec{A} + \vec{b} \)

### Step 3: Substitute for \( \vec{B} \) and \( \vec{C} \)
Now substituting for \( \vec{B} \) and \( \vec{C} \) in the equation for \( \vec{D} \):
\[ \vec{D} = \frac{1}{2} \left( (\vec{A} + \vec{c}) + (\vec{A} + \vec{b}) \right) \]
This simplifies to:
\[ \vec{D} = \frac{1}{2} (2\vec{A} + \vec{b} + \vec{c}) = \vec{A} + \frac{1}{2} (\vec{b} + \vec{c}) \]

### Step 4: Find \( \vec{AD} \)
Now, we can find the vector \( \vec{AD} \):
\[ \vec{AD} = \vec{D} - \vec{A} \]
Substituting \( \vec{D} \):
\[ \vec{AD} = \left( \vec{A} + \frac{1}{2} (\vec{b} + \vec{c}) \right) - \vec{A} = \frac{1}{2} (\vec{b} + \vec{c}) \]

### Step 5: Compare with Given Options
Now we need to compare \( \vec{AD} = \frac{1}{2} (\vec{b} + \vec{c}) \) with the given options:
1. \( \frac{1}{2} \vec{a} + \vec{c} \)
2. \( -\frac{1}{2} \vec{b} + \frac{1}{2} \vec{c} \)
3. \( \frac{1}{2} \vec{a} + \vec{b} \)

None of the options directly match \( \frac{1}{2} (\vec{b} + \vec{c}) \). However, we can manipulate the second option:
\[ -\frac{1}{2} \vec{b} + \frac{1}{2} \vec{c} = \frac{1}{2} \vec{c} - \frac{1}{2} \vec{b} \]
This can be rewritten as:
\[ \frac{1}{2} (\vec{c} - \vec{b}) \]
This does not match either.

### Conclusion
After analyzing the options, we find that none of the provided options correctly represent the median \( \vec{AD} \). Thus, the correct answer is that none of the options are correct.