Let `veca, vecb, vecc` be any three vectors.Then vectors `vecu=vecaxx(vecbxxvecc), vecv=vecbxx(veccxxveca)` and `vecw=veccxx(vecaxxvecb)` are such that they are

Options

A. collinear
B. non-coplanar
C. coplanar
D. none of these

Explanation

To solve the problem, we need to analyze the vectors \(\vec{u}\), \(\vec{v}\), and \(\vec{w}\) defined in terms of the cross products of the vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\).

### Step-by-Step Solution:

1. **Define the vectors**:
\[ \vec{u} = \vec{a} \times (\vec{b} \times \vec{c}) \]
\[ \vec{v} = \vec{b} \times (\vec{c} \times \vec{a}) \]
\[ \vec{w} = \vec{c} \times (\vec{a} \times \vec{b}) \]

2. **Use the vector triple product identity**:
The vector triple product identity states that:
\[ \vec{x} \times (\vec{y} \times \vec{z}) = (\vec{x} \cdot \vec{z}) \vec{y} - (\vec{x} \cdot \vec{y}) \vec{z} \]
We will apply this identity to each of the vectors \(\vec{u}\), \(\vec{v}\), and \(\vec{w}\).

3. **Calculate \(\vec{u}\)**:
\[ \vec{u} = \vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c} \]

4. **Calculate \(\vec{v}\)**:
\[ \vec{v} = \vec{b} \times (\vec{c} \times \vec{a}) = (\vec{b} \cdot \vec{a}) \vec{c} - (\vec{b} \cdot \vec{c}) \vec{a} \]

5. **Calculate \(\vec{w}\)**:
\[ \vec{w} = \vec{c} \times (\vec{a} \times \vec{b}) = (\vec{c} \cdot \vec{b}) \vec{a} - (\vec{c} \cdot \vec{a}) \vec{b} \]

6. **Add the vectors \(\vec{u}\), \(\vec{v}\), and \(\vec{w}\)**:
\[ \vec{u} + \vec{v} + \vec{w} = \left[(\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c}\right] + \left[(\vec{b} \cdot \vec{a}) \vec{c} - (\vec{b} \cdot \vec{c}) \vec{a}\right] + \left[(\vec{c} \cdot \vec{b}) \vec{a} - (\vec{c} \cdot \vec{a}) \vec{b}\right] \]

7. **Combine like terms**:
- The terms involving \(\vec{a}\):
\[ - (\vec{b} \cdot \vec{c}) \vec{a} + (\vec{c} \cdot \vec{b}) \vec{a} = 0 \]
- The terms involving \(\vec{b}\):
\[ (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{c} \cdot \vec{a}) \vec{b} = 0 \]
- The terms involving \(\vec{c}\):
\[ (\vec{b} \cdot \vec{a}) \vec{c} - (\vec{a} \cdot \vec{b}) \vec{c} = 0 \]

8. **Conclusion**:
Since all terms cancel out, we have:
\[ \vec{u} + \vec{v} + \vec{w} = \vec{0} \]
This implies that the vectors \(\vec{u}\), \(\vec{v}\), and \(\vec{w}\) are coplanar.

### Final Answer:
The vectors \(\vec{u}\), \(\vec{v}\), and \(\vec{w}\) are coplanar.