# Is right hand rule arbitrary?

In definition of vector multiplication, the direction of the resulting vector is given by the right hand rule. However I don't know any mathematical requirement to pick right hand instead of left for this purpose. Is there any?

• Is the right-handedness definition of vector math arbitrary? Yes. – NeutronStar Feb 3 '17 at 18:26
• Thanks. As a follow up question, why the right hand? – imcakmak Feb 3 '17 at 18:28
• Anecdotally, Eugene Wigner used the left hand rule for his calculations, converting as appropriate for the final answer. But then, he did know a little about group theory and symmetries and how to convert them in his head... – Jon Custer Feb 3 '17 at 19:30
• @Floris Thank you. In this ( evl.uic.edu/ralph/508S98/coordinates.html ) page, I see the fundamental difference between a right-handed coordinate system and a left-handed one is the direction of positive rotation. It is cw for a left-handed cor.sys. and ccw for a right-handed cs – imcakmak Feb 7 '17 at 18:03
• @user144302 - I always found the "volume concept" of the dot and cross products helpful. The volume of an object formed by vectors A,B,C is $(A\times B)\cdot C$. Nice explanation is given on the mathexchange site where this question was asked before... – Floris Feb 7 '17 at 20:01

You could define it with the left hand, too. In this case, the formula of the 3d vectorial multiplication would simply negated.

A physics could be also constructed for that, it would be exactly the same, of course the formulas using vectorial multiplication would be negated.

It is like a binary Higgs-mechanism, similarly as the analog, mechanical clocks are rotating to right. They could rotate also left. Some hundreds of years ago, a lot of them did.

• Sundials, south of the equator, still do go CCW. – Whit3rd Feb 3 '17 at 23:33
• @Whit3rd It is because the world would work exactly as before, only our formulas would change. – peterh - Reinstate Monica Jun 24 '17 at 7:40

Draw a 3D set of axis for XYZ. Where is positive X? Where is positive Y? Where is positive Z? Obviously wherever you want them but right, up and out of the page are the commonly accepted choices.

Now accepting that how do I remember it? I close the fingers of my right hand from X to Y and my thumb points to Z. If I instead preferred opening my fingers from X to Y I would use need to use my left hand.

Also this lets teachers be pretty sure who is going to fail during tests.

EDIT: The relation of the orientations is part of the definition of a cross product. We want to avoid adding negative signs if we don't have to. Using the traditional XYZ this means closing your right hand, but each time you flip the sign of an axis you can add a switch of hands to preserve a "stay positive" property.

• I understand how x,y and z are arbitrarily picked when representing on paper. However there could still be a requirement for a certain directional relationship between vectors AxB=C. And thats what I was asking. Turns out there isn't any, with the help of previous answers. – imcakmak Feb 4 '17 at 21:36