Take coordinate axes as follows. The x- and y-axes are both in the inclined plane (that's the key thing), with the x-axis horizontal and the y-axis up the line of greatest slope. The z-axis will then be perpendicular to the plane, so it will not be vertical. In fact, a vertical line will be in the direction of the vector .
If Arthur is at the point A and Bertha is at B, then the line joining O to the mid-point of AB will make an angle with the y-axis, and it's easy to see from this that the line AB will be in the direction . Now all you have to do is to take the inner product of this vector with the one in the vertical direction.