Let O be the origin and P, Q, R be the points such that `vec(PO) + vec(OQ) = vec(QO) + vec(OR)`. Then which one of the following is correct?

Options

A. P, Q, R are the vertices of an equilateral triangle
B. P, Q, R are the vertices of an equilateral triangle
C. P, Q, R are collinear
D. None of the above

Answer

To solve the problem, we start with the given equation:

\[ \vec{PO} + \vec{OQ} = \vec{QO} + \vec{OR} \]

### Step 1: Rewrite the vectors in terms of the origin
Since \(O\) is the origin, we can express the vectors as follows:
- \(\vec{PO} = -\vec{OP}\)
- \(\vec{OQ} = \vec{OQ}\)
- \(\vec{QO} = -\vec{OQ}\)
- \(\vec{OR} = \vec{OR}\)

Substituting these into the equation gives us:

\[ -\vec{OP} + \vec{OQ} = -\vec{OQ} + \vec{OR} \]

### Step 2: Rearranging the equation
Now, we can rearrange the equation:

\[ -\vec{OP} + \vec{OQ} + \vec{OQ} = \vec{OR} \]

This simplifies to:

\[ -\vec{OP} + 2\vec{OQ} = \vec{OR} \]

### Step 3: Isolate \(\vec{OP}\)
Rearranging further, we can isolate \(\vec{OP}\):

\[ \vec{OP} = 2\vec{OQ} - \vec{OR} \]

### Step 4: Express in terms of vectors
Now, we can express this in terms of the vectors from \(O\) to the points \(P\), \(Q\), and \(R\):

Let:
- \(\vec{OP} = \vec{p}\)
- \(\vec{OQ} = \vec{q}\)
- \(\vec{OR} = \vec{r}\)

So we have:

\[ \vec{p} = 2\vec{q} - \vec{r} \]

### Step 5: Analyze the result
This equation indicates a linear relationship among the points \(P\), \(Q\), and \(R\). Specifically, it shows that point \(P\) can be expressed as a linear combination of points \(Q\) and \(R\), which implies that points \(P\), \(Q\), and \(R\) are collinear.

### Conclusion
Thus, the correct answer is that points \(P\), \(Q\), and \(R\) are collinear.