The unit mass has `r = 8 hat(i) - 4 hat(j) and v = 8 hat(i) + 4 hat(j)` . Its angular momentum is

Options

A. 64 units in `-hat(k)` direction
B. 64 units in `+hat(k)` direction
C. 64 units in `+ hat(j)` direction
D. 64 units in `+ hat(i)` direction

Explanation

To find the angular momentum of a unit mass with the given position vector \( \mathbf{r} \) and velocity vector \( \mathbf{v} \), we can follow these steps:

### Step 1: Identify the position and velocity vectors
The position vector \( \mathbf{r} \) is given as:
\[ \mathbf{r} = 8 \hat{i} - 4 \hat{j} \]
The velocity vector \( \mathbf{v} \) is given as:
\[ \mathbf{v} = 8 \hat{i} + 4 \hat{j} \]

### Step 2: Calculate the linear momentum
The linear momentum \( \mathbf{P} \) is given by the product of mass and velocity. Since we have a unit mass (mass = 1), the linear momentum is equal to the velocity vector:
\[ \mathbf{P} = m \mathbf{v} = 1 \cdot (8 \hat{i} + 4 \hat{j}) = 8 \hat{i} + 4 \hat{j} \]

### Step 3: Use the formula for angular momentum
The angular momentum \( \mathbf{L} \) is calculated using the cross product of the position vector \( \mathbf{r} \) and the linear momentum \( \mathbf{P} \):
\[ \mathbf{L} = \mathbf{r} \times \mathbf{P} \]

### Step 4: Set up the cross product in matrix form
We can represent the cross product using a determinant:
\[ \mathbf{L} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 8 & -4 & 0 \\ 8 & 4 & 0 \end{vmatrix} \]

### Step 5: Calculate the determinant
Calculating the determinant, we have:
\[ \mathbf{L} = \hat{i} \begin{vmatrix} -4 & 0 \\ 4 & 0 \end{vmatrix} - \hat{j} \begin{vmatrix} 8 & 0 \\ 8 & 0 \end{vmatrix} + \hat{k} \begin{vmatrix} 8 & -4 \\ 8 & 4 \end{vmatrix} \]

Calculating each of these 2x2 determinants:
1. For \( \hat{i} \):
\[ \begin{vmatrix} -4 & 0 \\ 4 & 0 \end{vmatrix} = (-4)(0) - (0)(4) = 0 \]
2. For \( \hat{j} \):
\[ \begin{vmatrix} 8 & 0 \\ 8 & 0 \end{vmatrix} = (8)(0) - (0)(8) = 0 \]
3. For \( \hat{k} \):
\[ \begin{vmatrix} 8 & -4 \\ 8 & 4 \end{vmatrix} = (8)(4) - (-4)(8) = 32 + 32 = 64 \]

Putting it all together, we have:
\[ \mathbf{L} = 0 \hat{i} - 0 \hat{j} + 64 \hat{k} = 64 \hat{k} \]

### Step 6: Final Result
Thus, the angular momentum \( \mathbf{L} \) is:
\[ \mathbf{L} = 64 \hat{k} \text{ units} \]

### Conclusion
The correct answer is that the angular momentum is \( 64 \) units in the positive \( k \) direction.

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