For any three vectors `veca, vecb, vecc` the vector `(vecbxxvecc)xxveca` equals

Options

A. `(veca.vecb)vecc-(vecb.vecc)veca`
B. `(veca.vecb)vecc-(veca.vecc)vecb`
C. `(vecb.veca)vecc-(vecc.veca)vecb`
D. none of these

Explanation

To solve the problem, we need to find the expression for the vector \((\vec{b} \times \vec{c}) \times \vec{a}\) using the vector triple product identity. The vector triple product identity states that:

\[ \vec{x} \times (\vec{y} \times \vec{z}) = (\vec{x} \cdot \vec{z}) \vec{y} - (\vec{x} \cdot \vec{y}) \vec{z} \]

In our case, we can let \(\vec{x} = \vec{a}\), \(\vec{y} = \vec{b}\), and \(\vec{z} = \vec{c}\). So, we can rewrite our expression as follows:

### Step 1: Identify the vectors
We have:
- \(\vec{x} = \vec{a}\)
- \(\vec{y} = \vec{b}\)
- \(\vec{z} = \vec{c}\)

### Step 2: Apply the vector triple product identity
Using the vector triple product identity, we can express \((\vec{b} \times \vec{c}) \times \vec{a}\) as:

\[ (\vec{b} \times \vec{c}) \times \vec{a} = (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c} \]

### Step 3: Write the final expression
Thus, the final expression for \((\vec{b} \times \vec{c}) \times \vec{a}\) is:

\[ (\vec{b} \times \vec{c}) \times \vec{a} = (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c} \]

### Conclusion
The expression for \((\vec{b} \times \vec{c}) \times \vec{a}\) is \((\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c}\).

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