 Solution

Let $L$ be the fixed line and choose $y\in L$. Take orthonormal vectors $e_1,e_2$ spanning the plane perpendicular to $L$. The processis standard planar Brownian motion starting from $Y_0\ne 0$. Indeed the projected increments are centered Gaussian random vectors with covariance $(t-s)I_2$, are independent on disjoint intervals, and have continuous paths.

The event $B_t\in L$ is exactly $Y_t=0$. The polar point for planar Brownian motion result in (b) therefore givesThis is a projection criterion for Brownian avoidance of affine subspaces: avoiding a fixed point in a two-dimensional projection excludes hitting the original line.