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

Options

A. `lamda`
B. `sqrt3 lamda`
C. `sqrt5 lamda`
D. None of these

Answer

To solve the problem step by step, we will follow the vector operations involved in calculating the torque (moment) produced by a force acting on a spacecraft.

### Step 1: Identify the position vector and force vector
The position vector \( \mathbf{r} \) of the spacecraft is given as:
\[ \mathbf{r} = \hat{i} + 2\hat{j} + 3\hat{k} \]
The force vector \( \mathbf{F} \) applied by the rocket is given as:
\[ \mathbf{F} = \lambda \hat{k} \]

### Step 2: Write the vectors in component form
The position vector \( \mathbf{r} \) can be expressed in component form as:
\[ \mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} \]
The force vector \( \mathbf{F} \) can be expressed in component form as:
\[ \mathbf{F} = \begin{pmatrix} 0 \\ 0 \\ \lambda \end{pmatrix} \]

### Step 3: Calculate the torque using the cross product
The torque \( \mathbf{\tau} \) is calculated using the cross product of the position vector \( \mathbf{r} \) and the force vector \( \mathbf{F} \):
\[ \mathbf{\tau} = \mathbf{r} \times \mathbf{F} \]

Using the determinant method for the cross product:
\[ \mathbf{\tau} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 2 & 3 \\ 0 & 0 & \lambda \end{vmatrix} \]

### Step 4: Calculate the determinant
Calculating the determinant, we have:
\[ \mathbf{\tau} = \hat{i} \begin{vmatrix} 2 & 3 \\ 0 & \lambda \end{vmatrix} - \hat{j} \begin{vmatrix} 1 & 3 \\ 0 & \lambda \end{vmatrix} + \hat{k} \begin{vmatrix} 1 & 2 \\ 0 & 0 \end{vmatrix} \]

Calculating each of these 2x2 determinants:
1. \( \begin{vmatrix} 2 & 3 \\ 0 & \lambda \end{vmatrix} = 2\lambda - 0 = 2\lambda \)
2. \( \begin{vmatrix} 1 & 3 \\ 0 & \lambda \end{vmatrix} = 1\lambda - 0 = \lambda \)
3. \( \begin{vmatrix} 1 & 2 \\ 0 & 0 \end{vmatrix} = 0 \)

Thus, substituting back, we get:
\[ \mathbf{\tau} = 2\lambda \hat{i} - \lambda \hat{j} + 0 \hat{k} \]
So,
\[ \mathbf{\tau} = 2\lambda \hat{i} - \lambda \hat{j} \]

### Step 5: Calculate the magnitude of the torque
To find the magnitude of the torque \( |\mathbf{\tau}| \):
\[ |\mathbf{\tau}| = \sqrt{(2\lambda)^2 + (-\lambda)^2 + 0^2} \]
Calculating this gives:
\[ |\mathbf{\tau}| = \sqrt{4\lambda^2 + \lambda^2} = \sqrt{5\lambda^2} = \sqrt{5} \cdot |\lambda| \]

### Final Answer
Thus, the magnitude of the torque is:
\[ |\mathbf{\tau}| = \sqrt{5} \lambda \]