To determine whether the vectors \( \mathbf{a} \times \mathbf{b} \), \( \mathbf{b} \times \mathbf{c} \), and \( \mathbf{c} \times \mathbf{a} \) are coplanar or non-coplanar given that the vectors \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \) are non-coplanar, we can use the concept of scalar triple product.
### Step-by-Step Solution:
1. **Understanding Non-Coplanarity**:
Since the vectors \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \) are non-coplanar, the scalar triple product \( [\mathbf{a}, \mathbf{b}, \mathbf{c}] \) is non-zero. This means that the volume formed by these three vectors is not zero.
**Hint**: Remember that non-coplanar vectors form a three-dimensional volume.
2. **Forming Scalar Triple Products**:
We need to evaluate the scalar triple product of the vectors \( \mathbf{a} \times \mathbf{b} \), \( \mathbf{b} \times \mathbf{c} \), and \( \mathbf{c} \times \mathbf{a} \). The condition for coplanarity of three vectors \( \mathbf{u}, \mathbf{v}, \mathbf{w} \) is given by \( [\mathbf{u}, \mathbf{v}, \mathbf{w}] = \mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) = 0 \).
3. **Calculating the Scalar Triple Product**:
We can express the scalar triple product as:
\[
[\mathbf{a} \times \mathbf{b}, \mathbf{b} \times \mathbf{c}, \mathbf{c} \times \mathbf{a}] = (\mathbf{a} \times \mathbf{b}) \cdot ((\mathbf{b} \times \mathbf{c}) \times (\mathbf{c} \times \mathbf{a}))
\]
4. **Using Vector Triple Product Identity**:
We can apply the vector triple product identity:
\[
\mathbf{x} \times (\mathbf{y} \times \mathbf{z}) = (\mathbf{x} \cdot \mathbf{z}) \mathbf{y} - (\mathbf{x} \cdot \mathbf{y}) \mathbf{z}
\]
Let \( \mathbf{p} = \mathbf{b} \times \mathbf{c} \). Then:
\[
(\mathbf{b} \times \mathbf{c}) \times (\mathbf{c} \times \mathbf{a}) = (\mathbf{b} \cdot \mathbf{a}) \mathbf{c} - (\mathbf{c} \cdot \mathbf{a}) \mathbf{b}
\]
5. **Substituting Back**:
Substitute this back into our expression:
\[
[\mathbf{a} \times \mathbf{b}, \mathbf{b} \times \mathbf{c}, \mathbf{c} \times \mathbf{a}] = (\mathbf{a} \times \mathbf{b}) \cdot ((\mathbf{b} \cdot \mathbf{a}) \mathbf{c} - (\mathbf{c} \cdot \mathbf{a}) \mathbf{b})
\]
6. **Evaluating the Result**:
Since \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \) are non-coplanar, the scalar triple product \( [\mathbf{a}, \mathbf{b}, \mathbf{c}] \) is non-zero. Therefore, the resulting expression from the above step will also be non-zero.
7. **Conclusion**:
Since the scalar triple product \( [\mathbf{a} \times \mathbf{b}, \mathbf{b} \times \mathbf{c}, \mathbf{c} \times \mathbf{a}] \) is non-zero, the vectors \( \mathbf{a} \times \mathbf{b} \), \( \mathbf{b} \times \mathbf{c} \), and \( \mathbf{c} \times \mathbf{a} \) are non-coplanar.
### Final Answer:
The vectors \( \mathbf{a} \times \mathbf{b} \), \( \mathbf{b} \times \mathbf{c} \), and \( \mathbf{c} \times \mathbf{a} \) are **non-coplanar**.
