# Vector interpretation of Kepler's 2nd law ( r X a = 0 )

I just read the vector interpretation of Kepler's second law and the conclusion put me in a confusion. The interpretation concludes by demonstrating that r X a = 0, where boldfaced r and a are respectively position vector of the planet from the sun and the acceleration of the movement. The demonstration is interpretated as r and a being parallel to each other, which I understand but in the opposite direction and thus acceleration is directed towards the center is what I don't. How can we deduce whether any two vectors are parallel in the same or the opposite direction just by analyzing the cross product?

• We cannot deduce whether any two vectors are parallel in the same or the opposite direction just by analyzing the cross product. For another example, if ab=0 and a,b are real, then we cannot deduce that a=0 or b=0 or both are zero. Commented Apr 20, 2016 at 9:05
• If it is so then how is it interpreted that the acceleration acts in the direction opposite to the position vector? Commented Apr 20, 2016 at 9:18
• By Newton gravity law. Commented Apr 20, 2016 at 9:34
• Kepler Second Law, that "the line joining a planet and the Sun sweeps out equal areas during equal intervals of time", is this the same a geometric interpretation of Newton's conclusion that under a central force the motion is on a plane with constant angular momentum. So there is no need to "interpret this interpretation" by the trivial result that the vector product of two collinear vectors is zero. Commented Apr 20, 2016 at 13:12