Physics Question from JEE Main 2025 · 22 January, Shift 1

The position vectors of two $1 \, \text{kg}$ particles, $(A)$ and $(B)$, are given by $\vec{r}_A = \left(\alpha_1 t^2 \hat{i} + \alpha_2 t \hat{j} + \alpha_3 t \hat{k}\right) m$ and $\vec{r}_B = \left(\beta_1 t \hat{i} + \beta_2 t^2 \hat{j} + \beta_3 t \hat{k}\right) m$, respectively; $\left(\alpha_1 = 1 \, m/s^2, \; \alpha_2 = 3n \, m/s, \; \alpha_3 = 2 \, m/s, \; \beta_1 = 2 \, m/s, \; \beta_2 = -1 \, m/s^2, \; \beta_3 = 4p \, m/s\right)$, where $t$ is time and $n$ and $p$ are constants. At $t = 1 \, s$, $|\vec{V}_A| = |\vec{V}_B|$ and the velocities $\vec{V}_A$ and $\vec{V}_B$ of the particles are orthogonal to each other. At $t = 1 \, s$, the magnitude of angular momentum of particle $(A)$ with respect to the position of particle $(B)$ is $\sqrt{L} \; kg\,m^2 s^{-1}$. The value of $L$ is _____.