Question #40825

Let $L$ be the normal line to the parabola $y^2 = 4x$ at the point $P(t^2, 2t)$, where $t > 0$. Let $\vec{u}$ be the vector parallel to $L$ such that $\vec{u} \cdot \hat{i} = 1$. Let $\vec{w}$ be the orthogonal projection of $\vec{u}$ onto the plane $x + y + z = 0$. If the volume of the tetrahedron formed by the origin $O$ and the points with position vectors $\vec{a} = \hat{i} + 4\hat{j} + \hat{k}$, $\vec{b} = 2\hat{i} + \hat{j} + \hat{k}$, and $\vec{w}$ is zero, find the length of the segment $OP$.