Three points A,B, and C have position vectors `-2veca+3vecb+5vecc, veca+2vecb+3vecc` and `7veca-vecc` with reference to an origin O. Answer the following questions?
Which of the following is true?

Options

A. `vec(AC)=2vec(AB)`
B. `vec(AC)=-3vec(AB)`
C. `vec(AC)=3vec(AB)`
D. None of these

Explanation

To solve the problem, we need to find the position vectors of points A, B, and C, and then calculate the vectors AB and AC to analyze their relationship.

### Step 1: Identify the position vectors
The position vectors of points A, B, and C are given as follows:
- Position vector of A: \(\vec{A} = -2\vec{a} + 3\vec{b} + 5\vec{c}\)
- Position vector of B: \(\vec{B} = \vec{a} + 2\vec{b} + 3\vec{c}\)
- Position vector of C: \(\vec{C} = 7\vec{a} - \vec{c}\)

### Step 2: Calculate the vector AB
The vector AB is calculated as:
\[ \vec{AB} = \vec{B} - \vec{A} \]
Substituting the position vectors:
\[ \vec{AB} = (\vec{a} + 2\vec{b} + 3\vec{c}) - (-2\vec{a} + 3\vec{b} + 5\vec{c}) \]
Simplifying this:
\[ \vec{AB} = \vec{a} + 2\vec{b} + 3\vec{c} + 2\vec{a} - 3\vec{b} - 5\vec{c} \]
\[ \vec{AB} = (1 + 2)\vec{a} + (2 - 3)\vec{b} + (3 - 5)\vec{c} \]
\[ \vec{AB} = 3\vec{a} - \vec{b} - 2\vec{c} \]

### Step 3: Calculate the vector AC
The vector AC is calculated as:
\[ \vec{AC} = \vec{C} - \vec{A} \]
Substituting the position vectors:
\[ \vec{AC} = (7\vec{a} - \vec{c}) - (-2\vec{a} + 3\vec{b} + 5\vec{c}) \]
Simplifying this:
\[ \vec{AC} = 7\vec{a} - \vec{c} + 2\vec{a} - 3\vec{b} - 5\vec{c} \]
\[ \vec{AC} = (7 + 2)\vec{a} - 3\vec{b} + (-1 - 5)\vec{c} \]
\[ \vec{AC} = 9\vec{a} - 3\vec{b} - 6\vec{c} \]

### Step 4: Establish the relationship between AC and AB
Now we can express \(\vec{AC}\) in terms of \(\vec{AB}\):
\[ \vec{AC} = 9\vec{a} - 3\vec{b} - 6\vec{c} \]
We can factor out 3 from \(\vec{AB}\):
\[ \vec{AB} = 3\vec{a} - \vec{b} - 2\vec{c} \]
Thus,
\[ \vec{AC} = 3(3\vec{a} - \vec{b} - 2\vec{c}) = 3\vec{AB} \]

### Conclusion
This implies that the vector AC is three times the vector AB. Therefore, the correct answer is:
\[ \vec{AC} = 3\vec{AB} \]

### Final Answer
The correct option is **C: AC = 3AB**.