Mathematics Question from JEE Main 2024 · 30 January, Shift 2

Let $L_1: \vec{r} = (\hat{i} - \hat{j} + 2\hat{k}) + \lambda(\hat{i} - \hat{j} + 2\hat{k}), \lambda \in \mathbb{R}, L_2: \vec{r} = (\hat{j} - \hat{k}) + \mu(3\hat{i} + \hat{j} + p\hat{k}), \mu \in \mathbb{R}, \text{ and } L_3: \vec{r} = \delta(\hat{i} + m\hat{j} - \hat{k}), \delta \in \mathbb{R}$ be three lines such that $L_1$ is perpendicular to $L_2$, and $L_3$ is perpendicular to both $L_1$ and $L_2$. Then the point which lies on $L_3$ is: