Here is a step-by-step guide on how to solve the problem.
Notice that the Runge-Lenz vector, which is a conserved vector for motion in a Coulombic potential, equals the sum of two vectors, one of which has a constant length and the other whose magnitude is proportional to the speed and is perpendicular to the velocity vector. Hence, the velocity can be expressed as the difference of a constant vector and a vector of constant magnitude, all rotated by a right angle. This means that the hodograph is a circle.
Each of the three velocities \vec v_1, \vec v_2, and \vec v_3 represents a point in the hodograph, and together they form a triangle inscribed in the circle.
The distances from the origin of these points are v_1, v_2, and v_3, respectively. Since each subsequent velocity is rotated by 90 degrees with respect to the previous one, the origin lies on one of the sides of the triangle, and \vec v_2 forms a height of the triangle drawn from that side. Hence, the area of the triangle can be easily found.
Based on the definition of eccentricity, one can see that the eccentricity is equal to the ratio of the distance of the circle’s center from the origin to the radius of the circle. The former can be found using Hint 3, while the latter can be found using Hint 2.
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