Consider a planet orbiting the Sun, and let { P }_{ 1 }, { P }_{ 2 }, { P }_{ 3 }, and { P }_{ 4 } be the planet’s position at four corresponding time instants t_{ 1 }, t_{ 2 }, t_{ 3 }, and t_{ 4 } such that t_{ 2 }-{ t }_{ 1 }={ t }_{ 4 }-{ t }_{ 3 }. Letting O denote the position of the Sun, determine the ratio between the areas of the orbital sectors P1OP2 and P3OP4. Hint: (1) The area of triangle OAB defined by the two planar vectors \overrightarrow { c } and \overrightarrow { d } as shown is given by Area(ABC)=\left| \overrightarrow { c } \times \overrightarrow { d } \right| (2) the solution of this problem is a demonstration of Kepler’s second law