A particle travels with a constant velocity parallel to the X...

Let the particle move along a line parallel to the $X$-axis with a constant velocity $\vec{v} = v \hat{i}$.

The position vector of the particle at any time $t$ is $\vec{r} = x \hat{i} + y \hat{j} + z \hat{k}$.

Since the particle moves parallel to the $X$-axis, its $y$ and $z$ coordinates remain constant.

The angular momentum of the particle relative to the origin is given by $\vec{L} = \vec{r} \times \vec{p}$, where $\vec{p} = m\vec{v}$ is the linear momentum.

$\vec{L} = (x \hat{i} + y \hat{j} + z \hat{k}) \times (mv \hat{i})$

$\vec{L} = m v (x (\hat{i} \times \hat{i}) + y (\hat{j} \times \hat{i}) + z (\hat{k} \times \hat{i}))$

$\vec{L} = m v (0 - y \hat{k} + z \hat{j})$

$\vec{L} = m z v \hat{j} - m y v \hat{k}$

Since $m$, $v$, $y$, and $z$ are all constants, the angular momentum $\vec{L}$ remains constant in both magnitude and direction.