Unfortunately, for most configurations there is no circular arc that joins the two points and has the required tangents. For that to be possible, the two perpendicular line segments must have the same length.
The simplest way to smoothly join the two line segments is, I think, to use a quadratic Bézier curve, that is, a parabolic arc. You already have two of the control points, and the third is simply the intersection of the extensions of the two given line segments (what you call “vectors”). This will only work if the two directed line segments point “toward” each other, of course. There are well-known standard algorithms for plotting quadratic Bézier curves that you should be able to find with a simple Internet search. One down side of using a quadratic Bézier is that it doesn’t reduce to a circular arc for the cases in which you could use one.
There are many other possibilities for joining the two line segments with a unique curve, such as higher-order Bézier curves, but unlike the quadratic Bézier, you don’t get a unique solution. With other choices, you’ll need to add some arbitrary constraints in order to get a visually satisfying result. For instance, an interesting possibility is to use an Euler spiral. This is the curve used to join segments of road that go in different directions. The computations involved are more complicated, but you can find a starting point in the Wikipedia article that I refer to. To use an Euler spiral for this application will require a couple of arbitrary parameters that roughly correspond to the speeds at which a vehicle would be entering and leaving the arc.