`[vec(a)vec(b)vec(c )]=[vec(b)vec(c )vec(a)]=[vec(c )vec(a)vec(b)]`.

Answer

To solve the problem given by the equation \([ \vec{a} \, \vec{b} \, \vec{c} ] = [ \vec{b} \, \vec{c} \, \vec{a} ] = [ \vec{c} \, \vec{a} \, \vec{b} ]\), we will analyze the determinants formed by these vectors.

### Step-by-step Solution:

1. **Understanding the Determinant**:
The notation \([ \vec{a} \, \vec{b} \, \vec{c} ]\) represents the scalar triple product, which can be calculated using the determinant of a matrix formed by the components of the vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\).

2. **Define the Vectors**:
Let:
\[ \vec{a} = (a_x, a_y, a_z), \quad \vec{b} = (b_x, b_y, b_z), \quad \vec{c} = (c_x, c_y, c_z) \]

3. **Set Up the Determinants**:
The scalar triple product can be expressed as:
\[ [ \vec{a} \, \vec{b} \, \vec{c} ] = \begin{vmatrix} a_x & a_y & a_z \\ b_x & b_y & b_z \\ c_x & c_y & c_z \end{vmatrix} \]
Similarly, we can write:
\[ [ \vec{b} \, \vec{c} \, \vec{a} ] = \begin{vmatrix} b_x & b_y & b_z \\ c_x & c_y & c_z \\ a_x & a_y & a_z \end{vmatrix} \]
and
\[ [ \vec{c} \, \vec{a} \, \vec{b} ] = \begin{vmatrix} c_x & c_y & c_z \\ a_x & a_y & a_z \\ b_x & b_y & b_z \end{vmatrix} \]

4. **Using Properties of Determinants**:
The properties of determinants tell us that swapping two rows changes the sign of the determinant. Therefore:
- Swapping the first and second rows in the determinant of \([ \vec{b} \, \vec{c} \, \vec{a} ]\) gives:
\[ [ \vec{b} \, \vec{c} \, \vec{a} ] = -[ \vec{c} \, \vec{b} \, \vec{a} ] \]
- Similarly, swapping the first and second rows in the determinant of \([ \vec{c} \, \vec{a} \, \vec{b} ]\) gives:
\[ [ \vec{c} \, \vec{a} \, \vec{b} ] = -[ \vec{a} \, \vec{c} \, \vec{b} ] \]

5. **Equating the Determinants**:
Since we have:
\[ [ \vec{a} \, \vec{b} \, \vec{c} ] = [ \vec{b} \, \vec{c} \, \vec{a} ] = [ \vec{c} \, \vec{a} \, \vec{b} ] \]
we can conclude that all three determinants are equal. Thus, if one of them is zero, all are zero, indicating that the vectors are coplanar.

6. **Conclusion**:
The given condition implies that the vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) are coplanar, which means that the scalar triple product is zero.