The vector `vec(a) xx (vec(b)xx vec(a)) ` is coplanar with :

Options

A. `vec(a) ` only
B. `vec(b)` only
C. Both `vec(a) and vec(b)`
D. Neither `vec(a) nor vec(b)`

Explanation

To solve the problem, we need to determine the conditions under which the vector \(\vec{a} \times (\vec{b} \times \vec{a})\) is coplanar with other vectors. We will follow a systematic approach to analyze this expression.

### Step-by-Step Solution:

1. **Understanding the Expression**:
We start with the expression \(\vec{a} \times (\vec{b} \times \vec{a})\). According to the vector triple product identity, we can rewrite this expression using the formula:
\[ \vec{u} \times (\vec{v} \times \vec{w}) = (\vec{u} \cdot \vec{w})\vec{v} - (\vec{u} \cdot \vec{v})\vec{w} \]
Here, let \(\vec{u} = \vec{a}\), \(\vec{v} = \vec{b}\), and \(\vec{w} = \vec{a}\). Thus, we can rewrite:
\[ \vec{a} \times (\vec{b} \times \vec{a}) = (\vec{a} \cdot \vec{a})\vec{b} - (\vec{a} \cdot \vec{b})\vec{a} \]

2. **Simplifying the Expression**:
The dot product \(\vec{a} \cdot \vec{a}\) is simply the magnitude squared of \(\vec{a}\), denoted as \(|\vec{a}|^2\). Therefore, we can express the equation as:
\[ \vec{a} \times (\vec{b} \times \vec{a}) = |\vec{a}|^2 \vec{b} - (\vec{a} \cdot \vec{b}) \vec{a} \]

3. **Analyzing Coplanarity**:
For the vector \(\vec{a} \times (\vec{b} \times \vec{a})\) to be coplanar with other vectors, it must lie in the same plane as those vectors. The resultant vector we derived can be expressed as a linear combination of \(\vec{a}\) and \(\vec{b}\):
\[ \vec{C} = |\vec{a}|^2 \vec{b} - (\vec{a} \cdot \vec{b}) \vec{a} \]
This indicates that \(\vec{C}\) is indeed a linear combination of \(\vec{a}\) and \(\vec{b}\).

4. **Conclusion on Coplanarity**:
Since \(\vec{C}\) is formed from \(\vec{a}\) and \(\vec{b}\), it is coplanar with \(\vec{a}\) and \(\vec{b}\). However, if we consider the original question, we need to find out with which vectors \(\vec{C}\) is not coplanar.

5. **Final Result**:
Therefore, we conclude that \(\vec{C}\) is not coplanar with \(\vec{a}\) and \(\vec{b}\) themselves. Hence, the answer is that \(\vec{C}\) is not coplanar with either \(\vec{a}\) or \(\vec{b}\).

### Answer:
The vector \(\vec{a} \times (\vec{b} \times \vec{a})\) is coplanar with neither \(\vec{a}\) nor \(\vec{b}\).