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Why do we use the right hand rule to determine the direction of the vector resulting from using the cross product? A resultant vector that was directed in the opposite direction would also be perpendicular to both vectors...

EDIT: If we cross a x b, using the right hand rule we use point our index finger in direction of a, the middle finger in direction of b, and we use the thumb to get the direction of c. From a mathematical point of view, why can't c point in the opposite direction to the thumb?

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    $\begingroup$ It is defined that way $\endgroup$ Apr 30 '16 at 16:49
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We have to pick one way to do it - either the right hand rule or the left hand rule. It's not important which one we use, as long as everyone uses the same one - just like which side of the road we drive on. We'd get the same eventual results if we all used the left hand rule. Someone chose the right hand rule over the left hand rule, and it became convention, so that's what we use.

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It is the way the cross product and the convention of the coordinate system is defined.

https://en.wikipedia.org/wiki/Right-hand_rule#Coordinates

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  • $\begingroup$ Why do we define it this way? Would it make a difference if it was defined differently? $\endgroup$
    – kyczawon
    Apr 30 '16 at 16:51
  • $\begingroup$ If it was defined the other way around, we would probably indeed use the left hand rule. $\endgroup$
    – flawr
    Apr 30 '16 at 16:52
  • $\begingroup$ I mean mathematically speaking. If we cross a x b, we use point our index finger in direction of a, the middle finger in direction of b, and we use the thumb to get the direction of c. From a mathematical point of view, why can't c point in the opposite direction to the thumb? $\endgroup$
    – kyczawon
    Apr 30 '16 at 16:56
  • $\begingroup$ @kyczawon We could have defined it that way, provided we also reversed all of a long list of commensurate conventions. A simple proof that this works is to do some 3-dimensional vector geometry in front of a mirror. $\endgroup$ Apr 30 '16 at 17:00
  • $\begingroup$ @kyczawon We could've defined cross product with $a\times b$ pointing the other way. The mathematical community has agreed, however, to use the convention which uses right-hand rule. $\endgroup$
    – Wojowu
    Apr 30 '16 at 17:00

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