Let `vecr` be a non - zero vector satisfying `vecr.veca = vecr.vecb =vecr.vecc =0` for given non- zero vectors `veca vecb and vecc`
Statement 1: `[ veca - vecb vecb - vecc vecc- veca] =0`
Statement 2: `[veca vecb vecc] =0`

Options

A. Both the statements are true and statement 2 is the correct explanation for statement 1.
B. Both statements are true but statement 2 is not the correct explanation for statement 1.
C. Statement 1 is true and Statement 2 is false
D. Statement 1 is false and Statement 2 is true.

Explanation

To solve the problem, we need to analyze the statements given the conditions on the vector \(\vec{r}\).

### Step-by-Step Solution:

1. **Understanding the Conditions**:
We are given that \(\vec{r} \cdot \vec{a} = 0\), \(\vec{r} \cdot \vec{b} = 0\), and \(\vec{r} \cdot \vec{c} = 0\). This means that the vector \(\vec{r}\) is perpendicular to the vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\).

**Hint**: Recall that if the dot product of two vectors is zero, it implies that the vectors are perpendicular to each other.

2. **Analyzing Statement 2**:
We need to check if the vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) are coplanar. The condition for coplanarity of three vectors is that their scalar triple product is zero:
\[ [\vec{a}, \vec{b}, \vec{c}] = \vec{a} \cdot (\vec{b} \times \vec{c}) = 0 \]
Since \(\vec{r}\) is perpendicular to all three vectors, we can conclude that \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) are indeed coplanar.

**Hint**: Use the scalar triple product to check for coplanarity. If the scalar triple product is zero, the vectors are coplanar.

3. **Analyzing Statement 1**:
The first statement is \([\vec{a} - \vec{b}, \vec{b} - \vec{c}, \vec{c} - \vec{a}] = 0\). This can be rewritten using the property of the scalar triple product:
\[ [\vec{u}, \vec{v}, \vec{w}] = 0 \text{ if } \vec{u}, \vec{v}, \vec{w} \text{ are coplanar.} \]
Here, \(\vec{u} = \vec{a} - \vec{b}\), \(\vec{v} = \vec{b} - \vec{c}\), and \(\vec{w} = \vec{c} - \vec{a}\). Since \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) are coplanar, it follows that \(\vec{u}\), \(\vec{v}\), and \(\vec{w}\) are also coplanar.

**Hint**: Remember that the difference of vectors does not affect their coplanarity. If the original vectors are coplanar, their differences will also be coplanar.

4. **Conclusion**:
Both statements are true:
- Statement 1: \([\vec{a} - \vec{b}, \vec{b} - \vec{c}, \vec{c} - \vec{a}] = 0\) is true because the vectors are coplanar.
- Statement 2: \([\vec{a}, \vec{b}, \vec{c}] = 0\) is true because \(\vec{r}\) being perpendicular to all three vectors implies they are coplanar.

Therefore, both statements are true.

### Final Answer:
Both Statement 1 and Statement 2 are true.