A sailboat is located at a point $4$ km away from a lighthouse. In relation to the lighthouse, this point is positioned at $40$ degree north relative to the east. The sailboat moves towards a point located $6$ km from the lighthouse, with an orientation of $60$ degree north relative to the west, still from the lighthouse. a$)$ What is its displacement? b$)$ During its movement, what was the shortest distance between the sailboat and the lighthouse?