The question feels to me like it should be in 3 dimensions, since it says "perpendicular to and intersects" and the "intersects" part would be trivial in 2d (and I think Bill Cook's answer covers the 2d case).
As in that answer, the direction of your given line is the vector $c$ and you need to find a direction vector $d$ for the new line, which should be perpendicular to $c$. The direction vector $d$ also has to lie in the plane determined by the known line and the known point. We can find the normal vector $n$ to that plane by taking the cross-product of the direction vector $c$ of the line and a vector from $B$ to any point on the line. Once we have $n$, $d$ must satisfy two conditions: $d\cdot c=0$ so that our new line is perpendicular to the existing line and $d\cdot n=0$ so that our new line lies in the same plane as the existing line and the given point. Having two equations is sufficient to solve for a 3d vector up to the length of that vector—that is, any $d$ that satisfies those two equations will be in the same direction as the line we want, and an equation for that line will be $\vec{r}=b+\lambda d$.