`bara times (barb times barc) +barb times (barc times bara)+ barc times (bara times barb)` equals

Options

A. `3[bara barb barc]`
B. `2[bara barb barc]`
C. `[bara barb barc]`
D. 0

Answer

To solve the problem \( \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) + \mathbf{b} \times (\mathbf{c} \times \mathbf{a}) + \mathbf{c} \times (\mathbf{a} \times \mathbf{b}) \), we will use the vector triple product identity, which states that:

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

### Step 1: Apply the vector triple product identity

We will apply this identity to each term in the expression.

1. For \( \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) \):
\[ \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{a} \cdot \mathbf{b}) \mathbf{c} \]

2. For \( \mathbf{b} \times (\mathbf{c} \times \mathbf{a}) \):
\[ \mathbf{b} \times (\mathbf{c} \times \mathbf{a}) = (\mathbf{b} \cdot \mathbf{a}) \mathbf{c} - (\mathbf{b} \cdot \mathbf{c}) \mathbf{a} \]

3. For \( \mathbf{c} \times (\mathbf{a} \times \mathbf{b}) \):
\[ \mathbf{c} \times (\mathbf{a} \times \mathbf{b}) = (\mathbf{c} \cdot \mathbf{b}) \mathbf{a} - (\mathbf{c} \cdot \mathbf{a}) \mathbf{b} \]

### Step 2: Combine all the results

Now, we can combine all these results together:
\[ \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) + \mathbf{b} \times (\mathbf{c} \times \mathbf{a}) + \mathbf{c} \times (\mathbf{a} \times \mathbf{b}) \]
Substituting the expressions we found:
\[ = \left[(\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{a} \cdot \mathbf{b}) \mathbf{c}\right] + \left[(\mathbf{b} \cdot \mathbf{a}) \mathbf{c} - (\mathbf{b} \cdot \mathbf{c}) \mathbf{a}\right] + \left[(\mathbf{c} \cdot \mathbf{b}) \mathbf{a} - (\mathbf{c} \cdot \mathbf{a}) \mathbf{b}\right] \]

### Step 3: Simplify the expression

Now, we will group the terms:
\[ = (\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{c} \cdot \mathbf{a}) \mathbf{b} + (\mathbf{b} \cdot \mathbf{a}) \mathbf{c} - (\mathbf{a} \cdot \mathbf{b}) \mathbf{c} + (\mathbf{c} \cdot \mathbf{b}) \mathbf{a} - (\mathbf{b} \cdot \mathbf{c}) \mathbf{a} \]

Notice that:
- The terms \( (\mathbf{a} \cdot \mathbf{c}) \mathbf{b} \) and \( -(\mathbf{c} \cdot \mathbf{a}) \mathbf{b} \) cancel out.
- The terms \( (\mathbf{b} \cdot \mathbf{a}) \mathbf{c} \) and \( -(\mathbf{a} \cdot \mathbf{b}) \mathbf{c} \) cancel out.
- The terms \( (\mathbf{c} \cdot \mathbf{b}) \mathbf{a} \) and \( -(\mathbf{b} \cdot \mathbf{c}) \mathbf{a} \) cancel out.

### Final Result

Since all terms cancel out, we conclude that:
\[ \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) + \mathbf{b} \times (\mathbf{c} \times \mathbf{a}) + \mathbf{c} \times (\mathbf{a} \times \mathbf{b}) = \mathbf{0} \]

### Answer
The final answer is \( \mathbf{0} \).