Explain the Cartesian components of the angular momentum of a particle.

(N/A) The angular momentum $\vec{l}$ of a particle is defined as the cross product of its position vector $\vec{r}$ and linear momentum $\vec{p}$:
$\vec{l} = \vec{r} \times \vec{p}$
In Cartesian coordinates,$\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ and $\vec{p} = p_x\hat{i} + p_y\hat{j} + p_z\hat{k}$.
Calculating the cross product using the determinant form:
$\vec{l} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ x & y & z \\ p_x & p_y & p_z \end{vmatrix}$
Expanding the determinant:
$\vec{l} = \hat{i}(yp_z - zp_y) + \hat{j}(zp_x - xp_z) + \hat{k}(xp_y - yp_x)$
Comparing this with $\vec{l} = l_x\hat{i} + l_y\hat{j} + l_z\hat{k}$,we identify the Cartesian components:
$l_x = yp_z - zp_y$
$l_y = zp_x - xp_z$
$l_z = xp_y - yp_x$
These represent the components of angular momentum along the $X$,$Y$,and $Z$ axes,respectively.