`veca, vecb and vecc` are three coplanar vectors such that `veca + vecb + vecc=0`. If three vectors `vecp, vecq and vecr` are parallel to `veca, vecb and vecc`, respectively, and have integral but different magnitudes, then among the following options, `|vecp +vecq + vecr|` can take a value equal to

Options

A. `1`
B. `0`
C. `sqrt3`
D. `2`

Explanation

To solve the problem step by step, we start with the given information about the vectors.

### Step 1: Understand the Given Vectors
We have three coplanar vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) such that:
\[ \vec{a} + \vec{b} + \vec{c} = 0 \]
This implies that \(\vec{c} = -(\vec{a} + \vec{b})\).

### Step 2: Define Parallel Vectors
We define three new vectors \(\vec{p}\), \(\vec{q}\), and \(\vec{r}\) which are parallel to \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) respectively. We can express them as:
\[ \vec{p} = k_1 \vec{a}, \quad \vec{q} = k_2 \vec{b}, \quad \vec{r} = k_3 \vec{c} \]
where \(k_1\), \(k_2\), and \(k_3\) are scalar multiples (magnitudes) of the vectors and are integers.

### Step 3: Substitute \(\vec{c}\)
Substituting \(\vec{c}\) in terms of \(\vec{a}\) and \(\vec{b}\):
\[ \vec{r} = k_3 (-\vec{a} - \vec{b}) = -k_3 \vec{a} - k_3 \vec{b} \]

### Step 4: Calculate \(\vec{p} + \vec{q} + \vec{r}\)
Now, we can find the sum of the vectors:
\[ \vec{p} + \vec{q} + \vec{r} = k_1 \vec{a} + k_2 \vec{b} - k_3 \vec{a} - k_3 \vec{b} \]
This simplifies to:
\[ (k_1 - k_3) \vec{a} + (k_2 - k_3) \vec{b} \]

### Step 5: Find the Magnitude
The magnitude of \(\vec{p} + \vec{q} + \vec{r}\) is given by:
\[ |\vec{p} + \vec{q} + \vec{r}| = |(k_1 - k_3) \vec{a} + (k_2 - k_3) \vec{b}| \]
Using the properties of magnitudes, we can express this as:
\[ |\vec{p} + \vec{q} + \vec{r}| = \sqrt{(k_1 - k_3)^2 |\vec{a}|^2 + (k_2 - k_3)^2 |\vec{b}|^2 + 2(k_1 - k_3)(k_2 - k_3) |\vec{a}||\vec{b}|\cos(\theta)} \]
where \(\theta\) is the angle between \(\vec{a}\) and \(\vec{b}\).

### Step 6: Integral Magnitudes
Since \(k_1\), \(k_2\), and \(k_3\) are integral and different, we can choose values for \(k_1\), \(k_2\), and \(k_3\) such that they are distinct integers.

### Step 7: Evaluate Possible Values
We can try different combinations of \(k_1\), \(k_2\), and \(k_3\) to find the possible values of \(|\vec{p} + \vec{q} + \vec{r}|\).

For example, if we take:
- \(k_1 = 3\)
- \(k_2 = 2\)
- \(k_3 = 1\)

Then:
\[ |\vec{p} + \vec{q} + \vec{r}| = |(3 - 1)\vec{a} + (2 - 1)\vec{b}| = |2\vec{a} + 1\vec{b}| \]
This will yield a specific magnitude based on the values of \(|\vec{a}|\) and \(|\vec{b}|\).

### Conclusion
Through various combinations, we find that the possible values of \(|\vec{p} + \vec{q} + \vec{r}|\) can yield certain results. After testing the options provided, we conclude that the value can be:
\[ \text{Option 3: } \sqrt{3} \]