`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, we need to evaluate the expression:

\[ \text{bara} \times (\text{barb} \times \text{barc}) + \text{barb} \times (\text{barc} \times \text{bara}) + \text{barc} \times (\text{bara} \times \text{barb}) \]

This expression involves the cross product of vectors. We can use the vector triple product identity, which states:

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

Let's apply this identity step by step to each term in the expression.

### Step 1: Evaluate \(\text{bara} \times (\text{barb} \times \text{barc})\)

Using the vector triple product identity:

\[ \text{bara} \times (\text{barb} \times \text{barc}) = (\text{bara} \cdot \text{barc}) \text{barb} - (\text{bara} \cdot \text{barb}) \text{barc} \]

### Step 2: Evaluate \(\text{barb} \times (\text{barc} \times \text{bara})\)

Again, applying the vector triple product identity:

\[ \text{barb} \times (\text{barc} \times \text{bara}) = (\text{barb} \cdot \text{bara}) \text{barc} - (\text{barb} \cdot \text{barc}) \text{bara} \]

### Step 3: Evaluate \(\text{barc} \times (\text{bara} \times \text{barb})\)

Applying the vector triple product identity one more time:

\[ \text{barc} \times (\text{bara} \times \text{barb}) = (\text{barc} \cdot \text{barb}) \text{bara} - (\text{barc} \cdot \text{bara}) \text{barb} \]

### Step 4: Combine all three results

Now, we combine the results from Steps 1, 2, and 3:

\[ \begin{align*} & \left[(\text{bara} \cdot \text{barc}) \text{barb} - (\text{bara} \cdot \text{barb}) \text{barc}\right] + \left[(\text{barb} \cdot \text{bara}) \text{barc} - (\text{barb} \cdot \text{barc}) \text{bara}\right] + \left[(\text{barc} \cdot \text{barb}) \text{bara} - (\text{barc} \cdot \text{bara}) \text{barb}\right] \end{align*} \]

### Step 5: Simplify the expression

Notice that the terms will cancel out:

- The \((\text{bara} \cdot \text{barc}) \text{barb}\) and \(-(\text{barc} \cdot \text{bara}) \text{barb}\) will cancel with the \((\text{barc} \cdot \text{barb}) \text{bara}\) and \(-(\text{barb} \cdot \text{bara}) \text{barc}\).

Thus, the entire expression simplifies to:

\[ 0 \]

### Final Answer:

The value of the expression is \(0\).

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