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. `2vec(OA) -3vec(OB) + vec(OC)=vec0`
B. `2vec(OA)+7vec(OB)+9vec(OC)=vec0`
C. `vec(OA)+vec(OB)+vec(OC)=vec0`
D. None of these

Explanation

To solve the problem, we need to analyze the position vectors of points A, B, and C given in the question. Let's denote the position vectors as follows:

- Position vector of point A: \( \vec{A} = -2\vec{a} + 3\vec{b} + 5\vec{c} \)
- Position vector of point B: \( \vec{B} = \vec{a} + 2\vec{b} + 3\vec{c} \)
- Position vector of point C: \( \vec{C} = 7\vec{a} - \vec{c} \)

Now, we need to check which of the given options is true. We will use the method of substituting these vectors into a linear combination to see if it results in the zero vector.

### Step 1: Set up the equation
We need to check the linear combination:
\[ 2\vec{A} - 3\vec{B} + 1\vec{C} = 0 \]

### Step 2: Substitute the position vectors
Substituting the values of \( \vec{A}, \vec{B}, \) and \( \vec{C} \):
\[ 2(-2\vec{a} + 3\vec{b} + 5\vec{c}) - 3(\vec{a} + 2\vec{b} + 3\vec{c}) + (7\vec{a} - \vec{c}) = 0 \]

### Step 3: Expand the equation
Now, we expand each term:
\[ 2(-2\vec{a}) + 2(3\vec{b}) + 2(5\vec{c}) - 3(\vec{a}) - 3(2\vec{b}) - 3(3\vec{c}) + 7\vec{a} - \vec{c} = 0 \]
This simplifies to:
\[ -4\vec{a} + 6\vec{b} + 10\vec{c} - 3\vec{a} - 6\vec{b} - 9\vec{c} + 7\vec{a} - \vec{c} = 0 \]

### Step 4: Combine like terms
Now, we combine the coefficients of \( \vec{a}, \vec{b}, \) and \( \vec{c} \):
\[ (-4\vec{a} - 3\vec{a} + 7\vec{a}) + (6\vec{b} - 6\vec{b}) + (10\vec{c} - 9\vec{c} - \vec{c}) = 0 \]
This results in:
\[ 0\vec{a} + 0\vec{b} + 0\vec{c} = 0 \]

### Step 5: Conclusion
Since the left-hand side equals the zero vector, we conclude that:
\[ 2\vec{A} - 3\vec{B} + 1\vec{C} = 0 \]
This means the relation holds true, confirming that the statement is valid.

### Final Answer
The correct option is the one that states:
\[ 2\vec{A} - 3\vec{B} + 1\vec{C} = 0 \]