A particle of mass `2 kg` located at the position `(hat i+ hat k) m` has a velocity `2(+ hat i- hat j + hat k) m//s`. Its angular momentum about `z-`axis in `kg-m^(2)//s` is :

Options

A. `+4`
B. `+8`
C. `-4`
D. `-8`

Explanation

To find the angular momentum of a particle about the z-axis, we can follow these steps:

### Step 1: Identify the given parameters
- Mass of the particle, \( m = 2 \, \text{kg} \)
- Position vector, \( \mathbf{r} = \hat{i} + \hat{k} \, \text{m} \)
- Velocity vector, \( \mathbf{v} = 2(\hat{i} - \hat{j} + \hat{k}) \, \text{m/s} = 2\hat{i} - 2\hat{j} + 2\hat{k} \)

### Step 2: Write the angular momentum formula
The angular momentum \( \mathbf{L} \) about the origin is given by:
\[ \mathbf{L} = \mathbf{r} \times (m \mathbf{v}) \]

### Step 3: Calculate \( m \mathbf{v} \)
First, we calculate \( m \mathbf{v} \):
\[ m \mathbf{v} = 2(2\hat{i} - 2\hat{j} + 2\hat{k}) = 4\hat{i} - 4\hat{j} + 4\hat{k} \]

### Step 4: Set up the cross product
Now we need to calculate the cross product \( \mathbf{r} \times (m \mathbf{v}) \):
\[ \mathbf{L} = (\hat{i} + \hat{k}) \times (4\hat{i} - 4\hat{j} + 4\hat{k}) \]

### Step 5: Write the vectors in matrix form
We can express the cross product using the determinant of a matrix:
\[ \mathbf{L} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 0 & 1 \\ 4 & -4 & 4 \end{vmatrix} \]

### Step 6: Calculate the determinant
Calculating the determinant:
\[ \mathbf{L} = \hat{i} \begin{vmatrix} 0 & 1 \\ -4 & 4 \end{vmatrix} - \hat{j} \begin{vmatrix} 1 & 1 \\ 4 & 4 \end{vmatrix} + \hat{k} \begin{vmatrix} 1 & 0 \\ 4 & -4 \end{vmatrix} \]

Calculating each of the 2x2 determinants:
1. \( \begin{vmatrix} 0 & 1 \\ -4 & 4 \end{vmatrix} = (0)(4) - (1)(-4) = 4 \)
2. \( \begin{vmatrix} 1 & 1 \\ 4 & 4 \end{vmatrix} = (1)(4) - (1)(4) = 0 \)
3. \( \begin{vmatrix} 1 & 0 \\ 4 & -4 \end{vmatrix} = (1)(-4) - (0)(4) = -4 \)

Putting it all together:
\[ \mathbf{L} = 4\hat{i} - 0\hat{j} - 4\hat{k} = 4\hat{i} - 4\hat{k} \]

### Step 7: Find the z-component of angular momentum
The z-component of angular momentum \( L_z \) is the coefficient of \( \hat{k} \):
\[ L_z = -4 \, \text{kg m}^2/\text{s} \]

### Final Answer
Thus, the angular momentum about the z-axis is:
\[ \boxed{-4} \, \text{kg m}^2/\text{s} \]

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