Question

A particle has the position vector $\vec{r} = \hat{i} - 2\hat{j} + \hat{k}$ and the linear momentum $\vec{p} = 2\hat{i} - \hat{j} + \hat{k}$. Its angular momentum about the origin is}

• A. $-\hat{i} + \hat{j} - 3\hat{k}$
• B. $-\hat{i} + \hat{j} + 3\hat{k}$
• C. $\hat{i} - \hat{j} + 3\hat{k}$
• D. $\hat{i} - \hat{j} - 5\hat{k}$
• E. $\hat{i} - \hat{j} + 5\hat{k}$

Hint:

Remember the order of the cross product matters ($\vec{r} \times \vec{p} \neq \vec{p} \times \vec{r}$). Reversing the order will result in a sign error for all components!

Answer 1

The Correct Option is B

Concept: Angular momentum ($\vec{L}$) of a particle about the origin is defined as the cross product of its position vector ($\vec{r}$) and its linear momentum ($\vec{p}$).
• Formula: $\vec{L} = \vec{r} \times \vec{p}$ 171].
• Determinant Method: The cross product is calculated using a $3 \times 3$ matrix.

Step 1: Set up the determinant for the cross product.
\[ \vec{L} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & -2 & 1 \\ 2 & -1 & 1 \end{vmatrix} \]

Step 2: Expand the determinant.
\[ \vec{L} = \hat{i}[(-2)(1) - (1)(-1)] - \hat{j}[(1)(1) - (1)(2)] + \hat{k}[(1)(-1) - (-2)(2)] \] \[ \vec{L} = \hat{i}[-2 + 1] - \hat{j}[1 - 2] + \hat{k}[-1 + 4] \] \[ \vec{L} = \hat{i}(-1) - \hat{j}(-1) + \hat{k}(3) \] \[ \vec{L} = -\hat{i} + \hat{j} + 3\hat{k} \]