If `veca, vecb` and `vecc` are unit coplanar vectors, then
`[(2veca-3vecb,7vecb-9vecc,12vecc-23veca)]`

Options

A. 0
B. `1/2`
C. `24`
D. `32`

Explanation

To solve the problem, we need to evaluate the expression given by the determinant of three vectors formed from the unit coplanar vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\). Let's break it down step by step.

### Step 1: Understand the properties of the vectors
Given that \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) are unit coplanar vectors, we know:
- \(|\vec{a}| = |\vec{b}| = |\vec{c}| = 1\) (they are unit vectors)
- The vectors are coplanar, which implies that the scalar triple product \(\vec{a} \cdot (\vec{b} \times \vec{c}) = 0\).

### Step 2: Set up the determinant
We need to evaluate the determinant formed by the vectors:
\[ \begin{vmatrix} 2\vec{a} - 3\vec{b} \\ 7\vec{b} - 9\vec{c} \\ 12\vec{c} - 23\vec{a} \end{vmatrix} \]

This can be represented in determinant form as:
\[ \begin{vmatrix} 2 & -3 & 0 \\ 0 & 7 & -9 \\ -23 & 0 & 12 \end{vmatrix} \]

### Step 3: Calculate the determinant
To calculate the determinant, we can use the formula for a \(3 \times 3\) determinant:
\[ \text{Det} = a(ei-fh) - b(di-fg) + c(dh-eg) \]
Where the matrix is:
\[ \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \]

For our matrix:
- \(a = 2\), \(b = -3\), \(c = 0\)
- \(d = 0\), \(e = 7\), \(f = -9\)
- \(g = -23\), \(h = 0\), \(i = 12\)

Calculating the determinant:
\[ \text{Det} = 2(7 \cdot 12 - (-9) \cdot 0) - (-3)(0 \cdot 12 - (-9)(-23)) + 0(0 \cdot 0 - 7 \cdot (-23)) \]
\[ = 2(84) + 3(207) + 0 \]
\[ = 168 + 621 \]
\[ = 789 \]

### Step 4: Consider the coplanarity condition
However, since \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) are coplanar, the determinant of any linear combination of these vectors should also equal zero. Therefore, the value of the determinant we calculated must be adjusted based on the coplanarity condition.

### Conclusion
Thus, the value of the expression is:
\[ \boxed{0} \]