A spacecraft at `hat(i)+2hat(j)+3hat(k)` is subjected to a force `lambda hat(k)` by firing a rocket. The spacecraft is subjected to a moment of magnitude

Options

A. `lambda`
B. `sqrt(3)lambda`
C. `sqrt(5)lambda`
D. None of these

Explanation

To solve the problem step by step, we will follow the outlined procedure to find the moment of the spacecraft subjected to the force.

### Step-by-Step Solution:

1. **Identify the Position Vector (r)**:
The position vector of the spacecraft is given as:
\[ \mathbf{r} = \hat{i} + 2\hat{j} + 3\hat{k} \]

2. **Identify the Force Vector (F)**:
The force vector acting on the spacecraft is given as:
\[ \mathbf{F} = \lambda \hat{k} \]

3. **Calculate the Moment (Torque) Vector (τ)**:
The moment (or torque) vector is calculated using the cross product of the position vector and the force vector:
\[ \mathbf{\tau} = \mathbf{r} \times \mathbf{F} \]

4. **Set Up the Determinant for the Cross Product**:
We will use the determinant method to compute the cross product:
\[ \mathbf{\tau} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 2 & 3 \\ 0 & 0 & \lambda \end{vmatrix} \]

5. **Calculate the Determinant**:
Expanding the determinant along the first row:
\[ \mathbf{\tau} = \hat{i} \begin{vmatrix} 2 & 3 \\ 0 & \lambda \end{vmatrix} - \hat{j} \begin{vmatrix} 1 & 3 \\ 0 & \lambda \end{vmatrix} + \hat{k} \begin{vmatrix} 1 & 2 \\ 0 & 0 \end{vmatrix} \]

Calculating each of the 2x2 determinants:
\[ = \hat{i} (2\lambda - 0) - \hat{j} (1\lambda - 0) + \hat{k} (0 - 0) \]
\[ = 2\lambda \hat{i} - \lambda \hat{j} + 0 \hat{k} \]
Therefore, we have:
\[ \mathbf{\tau} = 2\lambda \hat{i} - \lambda \hat{j} \]

6. **Find the Magnitude of the Moment (|τ|)**:
The magnitude of the moment vector is calculated as:
\[ |\mathbf{\tau}| = \sqrt{(2\lambda)^2 + (-\lambda)^2} \]
\[ = \sqrt{4\lambda^2 + \lambda^2} = \sqrt{5\lambda^2} = \sqrt{5} \cdot \lambda \]

7. **Final Answer**:
Thus, the magnitude of the moment is:
\[ |\mathbf{\tau}| = \sqrt{5} \lambda \]

### Conclusion:
The final answer is:
\[ \text{Magnitude of moment} = \sqrt{5} \lambda \]