If `u=hat(i)(atimeshat(i))+hat(j)(atimeshat(j))+hat(k)(atimeshat(k))`, then

Options

A. `u=0`
B. `u=hat(i)+hat(j)+hat(k)`
C. `u=2a`
D. `u=a`

Answer

To solve the problem, we need to analyze the expression given for the vector \( \mathbf{u} \):

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

### Step 1: Understand the Cross Product
The cross product of any vector with itself is zero. Hence, we have:
\[ \mathbf{a} \times \hat{i} = 0 \quad \text{(since } \hat{i} \times \hat{i} = 0\text{)} \]
\[ \mathbf{a} \times \hat{j} = 0 \quad \text{(since } \hat{j} \times \hat{j} = 0\text{)} \]
\[ \mathbf{a} \times \hat{k} = 0 \quad \text{(since } \hat{k} \times \hat{k} = 0\text{)} \]

### Step 2: Substitute the Cross Products
Now, substituting these results back into the expression for \( \mathbf{u} \):
\[ \mathbf{u} = \hat{i} (0) + \hat{j} (0) + \hat{k} (0) = \mathbf{0} \]

### Step 3: Conclusion
Thus, we find that:
\[ \mathbf{u} = \mathbf{0} \]

### Final Answer
The vector \( \mathbf{u} \) is equal to the zero vector.