the particle travels. The LRL vector A is perpendicular to the angular momentum vector L because both and r are perpendicular to L. It follows that A lies in the plane of motion. Alternative formulations for the same constant of motion may be defined, typically by scaling the vector with constants, such as the mass , the force parameter or the angular momentum . The most common variant is to divide A by , which yields the eccentricity vector, a dimensionless vector along the semi-major axis whose modulus equals the eccentricity of the conic: An equivalent formulation multiplies this eccentricity vector by the major semiaxis a, giving the resulting vector the units of length. Yet another formulation divides A by , yielding an equivalent conserved quantity with units of inverse length, a quantity that appears in the solution of the Kepler problem where is the angle between A and the position vector r. Further alternative formulations are given below. Derivation of the Kepler orbits The shape and orientation of the orbits can be determined from the LRL vector as follows. Taking the dot product of A with the position vector r gives the equation where θ is the angle between r and A (Figure 2). Permuting the scalar triple product yields Rearranging yields the solution for the Kepler equation This corresponds to the formula for a conic section of eccentricity e where the eccentricity and C is a constant. Taking the dot product of A with itself yields an equation involving the total energy , which may be rewritten in terms of the eccentricity, Thus, if the energy E is negative (bound orbits), the eccentricity is less than one and the orbit is an ellipse. Conversely, if the energy is positive (unbound orbits, also called "scattered orbits"), the eccentricity is greater than one and the orbit is a hyperbola. Finally, if the energy is exactly zero, the eccentricity is one and the orbit is a parabola. In all cases, the direction of A lies along the symmetry axis of the conic section and points from the center of force toward the periapsis, the point of closest approach. Circular momentum hodographs The conservation of the LRL vector A and angular momentum vector L is useful in showing that the momentum vector p moves on a circle under an inverse-square central force. Taking the dot product of with itself yields Further choosing