If ` veca xx (vecbxx vecc)= (veca xx vecb)xxvecc` then

Options

A. `(veccxxveca)xx vecb =vec0`
B. `vecc xx (vecaxxvecb)=vec0`
C. `vecbxx (vecc xxveca)=vec0`
D. `veccxxvecaxxvecb=vecb xx (veccxxveca) = vec0`

Explanation

To solve the problem, we need to show that if \( \vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \times \vec{b}) \times \vec{c} \), then the angle between \( \vec{a} \) and \( \vec{c} \) is \( 0^\circ \).

### Step-by-step Solution:

1. **Start with the given equation**:
\[ \vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \times \vec{b}) \times \vec{c} \]

2. **Use the vector triple product identity**:
The vector triple product identity states that:
\[ \vec{u} \times (\vec{v} \times \vec{w}) = (\vec{u} \cdot \vec{w}) \vec{v} - (\vec{u} \cdot \vec{v}) \vec{w} \]
Applying this to both sides:
- Left side:
\[ \vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c} \]
- Right side:
\[ (\vec{a} \times \vec{b}) \times \vec{c} = (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{b} \cdot \vec{c}) \vec{a} \]

3. **Set both sides equal**:
Now we have:
\[ (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c} = (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{b} \cdot \vec{c}) \vec{a} \]

4. **Rearranging the equation**:
By rearranging the equation, we can cancel out the common term \( (\vec{a} \cdot \vec{c}) \vec{b} \):
\[ - (\vec{a} \cdot \vec{b}) \vec{c} = - (\vec{b} \cdot \vec{c}) \vec{a} \]
Which simplifies to:
\[ (\vec{a} \cdot \vec{b}) \vec{c} = (\vec{b} \cdot \vec{c}) \vec{a} \]

5. **Assume \( \vec{c} \) is a scalar multiple of \( \vec{a} \)**:
Let \( \vec{c} = k \vec{a} \) for some scalar \( k \). Then substituting this back, we have:
\[ (\vec{a} \cdot \vec{b}) (k \vec{a}) = (\vec{b} \cdot (k \vec{a})) \vec{a} \]
This simplifies to:
\[ k (\vec{a} \cdot \vec{b}) \vec{a} = k (\vec{b} \cdot \vec{a}) \vec{a} \]

6. **Conclude the angle between \( \vec{a} \) and \( \vec{c} \)**:
Since \( \vec{c} = k \vec{a} \), the angle \( \theta \) between \( \vec{a} \) and \( \vec{c} \) is \( 0^\circ \) (they are in the same direction).

### Final Answer:
The angle between \( \vec{a} \) and \( \vec{c} \) is \( 0^\circ \).