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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?

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  • $\begingroup$ 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. $\endgroup$
    – lucas
    Apr 20, 2016 at 9:05
  • $\begingroup$ If it is so then how is it interpreted that the acceleration acts in the direction opposite to the position vector? $\endgroup$
    – Thomson1
    Apr 20, 2016 at 9:18
  • $\begingroup$ By Newton gravity law. $\endgroup$
    – lucas
    Apr 20, 2016 at 9:34
  • $\begingroup$ 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. $\endgroup$
    – Frobenius
    Apr 20, 2016 at 13:12

1 Answer 1

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The cross product of two vectors is the area of a parallelogram defined by those vectors. If the angle is 90 degrees then the cross product is simply the magnitude of vector 1 multiplied by the magnitude of vector 2 (i.e are of a rectangle). another way of writing the cross product is |v||u|sin(theta). If theta is 0 (both pointing in same direction) or 180 (opposite direction) then sin(theta) = 0. There is no area defined by them. Here you can see which area i'm talking about enter image description here

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  • $\begingroup$ thanks for the input. Yeah as I have written I understand that cross vector 0 means they have to be parallel. The only problem was I didn't understand how the author of the lesson that I read interpret that a is acting in the opposite direction (180 degrees) to r. Maybe as someone told he directly used Newton's Law. $\endgroup$
    – Thomson1
    Apr 20, 2016 at 10:09
  • $\begingroup$ ohh, well **a**=**F**/m The force vector is towards the sun so a is towards the sun. $\endgroup$
    – Nicolas
    Apr 20, 2016 at 10:39

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