Spherical trigonometry uses the known sides and angles of a spherical triangle formed by great circle arcs to find those quantities that are unknown. In the figure below for the English Channel problem the labels for the station points are also used for the interior angles, M, R and X, of the spherical triangle and the sides opposite to these angles, x, r and m, are also indicated.
The unit vector tangent to a great circle arc at a point can be used to find the bearing to some other point on the arc. The trick is to eliminate the component of the second vector in the direction of the first. The two simple procedures shown below are very useful. They determine the direction of the tangent and the heading from the first point to the second. The functions Er, Eθ and Eφ compute the radial and angular unit vectors from the angular coordinates of a point.
We started with the coordinates and headings for stations M and R. We need the heading of R relative to M to help determine the interior angles at M and R. The dot product of the two position vectors, eM and eR, are used to find the length of the connecting arc, x.
We now have enough information to calculate the interior angle at X using the law of cosines for angles. The values of X and x allow us to calculate the common ratio, ρ, for use with the law of sines and we can then find the lengths of sides r and m.
Either r or m can be used to find the position vector for station X. For m the formulas are,
eX = eM cos(m) + en sin(m).
E and N are the tangents to the sphere at M that give the direction of increase for φ and θ. The directions of the North Pole, the intersection of the Prime Meridian and the Equator and their easterly normal vector can be used to find the coordinates of station X.
The search that was done to find the intersection point wasn't really necessary. But it shows more directly that the arcs and their intersection are what determines the position for X.