If `u = i xx (a xx i), + j xx (a xx j) + k xx(a xx k)`, then

Options

A. u is a unit vector
B. `u = a + i+j+k`
C. `u = 2a`
D. `u=8(i+j+k)`

Answer

To solve the vector equation given in the question, we will use the vector triple product identity. The expression we have is:

\[ u = \mathbf{i} \times (\mathbf{a} \times \mathbf{i}) + \mathbf{j} \times (\mathbf{a} \times \mathbf{j}) + \mathbf{k} \times (\mathbf{a} \times \mathbf{k}) \]

### Step 1: Apply the Vector Triple Product Identity

The vector triple product identity states that:

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

We will apply this identity to each term in the expression for \( u \).

### Step 2: Calculate Each Component

1. **First Component**:
\[ \mathbf{i} \times (\mathbf{a} \times \mathbf{i}) = (\mathbf{i} \cdot \mathbf{i})\mathbf{a} - (\mathbf{i} \cdot \mathbf{a})\mathbf{i} \]
Since \( \mathbf{i} \cdot \mathbf{i} = 1 \):
\[ = \mathbf{a} - (\mathbf{a} \cdot \mathbf{i})\mathbf{i} \]

2. **Second Component**:
\[ \mathbf{j} \times (\mathbf{a} \times \mathbf{j}) = (\mathbf{j} \cdot \mathbf{j})\mathbf{a} - (\mathbf{j} \cdot \mathbf{a})\mathbf{j} \]
Since \( \mathbf{j} \cdot \mathbf{j} = 1 \):
\[ = \mathbf{a} - (\mathbf{a} \cdot \mathbf{j})\mathbf{j} \]

3. **Third Component**:
\[ \mathbf{k} \times (\mathbf{a} \times \mathbf{k}) = (\mathbf{k} \cdot \mathbf{k})\mathbf{a} - (\mathbf{k} \cdot \mathbf{a})\mathbf{k} \]
Since \( \mathbf{k} \cdot \mathbf{k} = 1 \):
\[ = \mathbf{a} - (\mathbf{a} \cdot \mathbf{k})\mathbf{k} \]

### Step 3: Combine All Components

Now, we can combine all three components:

\[ u = \left( \mathbf{a} - (\mathbf{a} \cdot \mathbf{i})\mathbf{i} \right) + \left( \mathbf{a} - (\mathbf{a} \cdot \mathbf{j})\mathbf{j} \right) + \left( \mathbf{a} - (\mathbf{a} \cdot \mathbf{k})\mathbf{k} \right) \]

This simplifies to:

\[ u = 3\mathbf{a} - \left( (\mathbf{a} \cdot \mathbf{i})\mathbf{i} + (\mathbf{a} \cdot \mathbf{j})\mathbf{j} + (\mathbf{a} \cdot \mathbf{k})\mathbf{k} \right) \]

### Step 4: Final Expression

Thus, the final expression for \( u \) is:

\[ u = 3\mathbf{a} - \mathbf{a} \]

This leads us to:

\[ u = 2\mathbf{a} \]