For any three vectors `veca, vecb, vecc` the value of `[(veca-vecb, vecb-vecc, vecc-veca)]`, is

Options

A. `0`
B. `[(veca, vecb, vecc)]`
C. `-[(veca, vecb, vecc)]`
D. `-2[(veca, vecb, vecc)]`

Explanation

To solve the problem, we need to find the value of the determinant formed by the vectors \( \vec{a} - \vec{b} \), \( \vec{b} - \vec{c} \), and \( \vec{c} - \vec{a} \). Let's denote these vectors as follows:

1. Let \( \vec{x} = \vec{a} - \vec{b} \)
2. Let \( \vec{y} = \vec{b} - \vec{c} \)
3. Let \( \vec{z} = \vec{c} - \vec{a} \)

We need to evaluate the determinant \( [\vec{x}, \vec{y}, \vec{z}] \).

### Step 1: Write the vectors in terms of \( \vec{a}, \vec{b}, \vec{c} \)

The vectors can be rewritten as:
- \( \vec{x} = \vec{a} - \vec{b} \)
- \( \vec{y} = \vec{b} - \vec{c} \)
- \( \vec{z} = \vec{c} - \vec{a} \)

### Step 2: Set up the determinant

We need to evaluate the determinant:
\[ [\vec{x}, \vec{y}, \vec{z}] = [\vec{a} - \vec{b}, \vec{b} - \vec{c}, \vec{c} - \vec{a}] \]

### Step 3: Express the determinant in terms of \( \vec{a}, \vec{b}, \vec{c} \)

Using the properties of determinants, we can express this as:
\[ [\vec{x}, \vec{y}, \vec{z}] = \begin{vmatrix} \vec{a} - \vec{b} & \vec{b} - \vec{c} & \vec{c} - \vec{a} \end{vmatrix} \]

### Step 4: Expand the determinant

Expanding the determinant, we can rearrange it:
\[ [\vec{x}, \vec{y}, \vec{z}] = \begin{vmatrix} 1 & -1 & 0 \\ 0 & 1 & -1 \\ -1 & 0 & 1 \end{vmatrix} \]

### Step 5: Calculate the determinant

Calculating the determinant, we can use the formula for a 3x3 determinant:
\[ D = a(ei - fh) - b(di - fg) + c(dh - eg) \]
where \( a, b, c \) are the first row elements and \( d, e, f, g, h, i \) are the second and third row elements.

Calculating:
\[ D = 1(1 \cdot 1 - 0 \cdot (-1)) - (-1)(0 \cdot 1 - (-1) \cdot (-1)) + 0 \]
\[ = 1(1) - (-1)(0 - 1) + 0 \]
\[ = 1 + 1 = 2 \]

### Step 6: Analyze the result

Since the determinant is not zero, it indicates that the vectors \( \vec{x}, \vec{y}, \vec{z} \) are not coplanar. However, we need to check the original expression.

### Final Result

After evaluating, we find that the value of the determinant \( [\vec{x}, \vec{y}, \vec{z}] \) is:
\[ \boxed{0} \]