Right hand rule in vector cross product

[vaishakh]

I have learned just about the right hand rule in vector cross product. How is this proved? Can anybody give an example where the cross product plays an important role and where the vector cross product formula is obeyed?
Our professor just told us that the torque due to a force acting on a body is towards us if it is rotating anti-clockwise and away from us if it is rotating clockwise. I just looked up at the fan in the classroom. It was rotating anticlockwise. I wonder whether the wind produced is due to the torque of the force that is being applied by the motor on the leaves. So the wind should be upward if the fan rotates clockwise. I sense some mistake. But I am not able to frame out.

 

[daveed]

Wouldn't the wind that comes from the fan depend on which way the fan blades tilted?

 

[berkeman]

I have learned just about the right hand rule in vector cross product. How is this proved? Can anybody give an example where the cross product plays an important role and where the vector cross product formula is obeyed?

Proved? Holy smokes. E&M? Transformers? The real world?

 

[James R]

I have learned just about the right hand rule in vector cross product. How is this proved?

Actually, it isn't. It's a somewhat arbitrary definition. We could just as easily use a left hand rule instead, and it would make no difference to any physical calculation, as long as we used it consistently.

Our professor just told us that the torque due to a force acting on a body is towards us if it is rotating anti-clockwise and away from us if it is rotating clockwise.

Sort of. If you curl the fingers of your right hand in the direction of rotation, then your thumb points in the direction of the angular velocity vector, or equivalently, angular momentum, as defined.

When you're working out the direction of a torque, curl your fingers in the direction the torque is trying to turn the object (even though the object might be turning in the opposite direction). Then your thumb points in the direction of the torque vector.

I just looked up at the fan in the classroom. It was rotating anticlockwise. I wonder whether the wind produced is due to the torque of the force that is being applied by the motor on the leaves.

The wind is produced by the force the fan blades exert on the air. The direction (up or down) is governed by the angle of the fan blades, not necessarily the rotation direction.

 

[rbj]

Actually, it isn't. It's a somewhat arbitrary definition. We could just as easily use a left hand rule instead, and it would make no difference to any physical calculation, as long as we used it consistently.

exactly. the term we would use for such an arbitrary definition vs. at least one other equally valid definition, that only works if we pick one and use it consistently, is "convention". the "right-hand rule" is a convention. we had to decide on that or the left-hand rule and, i s'pose because there are more right-handed physicists, they settled on the right hand rule.
it's sort of like the $\sqrt{-1}$. like any other square root, there are two equally plausible solutions. you find one solution and its negative is also a solution. since there is no real number that, when squared, becomes -1, we imagine a number and call it $i$. then we know that $-i$ is also a perfectly good solution. now, just because we picked $i$ for our imaginary unit, doesn't mean that the aliens on the planet Zog are wrong if they picked $-i$, as long as they are consistent. that means everywhere they see an $i$ in one of our textbooks, they'll plug in a $-i$ and everywhere they see a $-i$ in our textbooks, they'll replace it with $i$. so their Fourier Transform and many other definitions are different from ours, but just as valid, as long as they're consistent.
the right-hand rule (over the left-hand rule) is just as arbitrary.

 

[vaishakh]

I didn't quite understand your explanation evenafter rereading. please specify what is you point. my question is How is right hand rule of Vector cross product provedor are there experiments which could have led to it?

 

[Doc Al]

I think you're missing the point of the previous posters: The "right hand rule" is just a useful way of remembering the defined direction of the vector cross product. The vector cross product itself is a mathematical invention. Is the concept of vector cross product useful? You bet! It allows a concise mathematical description of many aspects of physics, for one.

If I missed your point, please restate it.

 

[NateTG]

I didn't quite understand your explanation evenafter rereading. please specify what is you point. my question is How is right hand rule of Vector cross product provedor are there experiments which could have led to it?

Let's say we have some object, and we want to describe how that object is spinning. Then we need to describe the axis that the object is spinning on, and how fast the object is spinning.

Now, for any particular axis, we can spin clockwise, and anticlockwise, so we'd like to say that one is spinning 'forwards' and the other one is 'backwards'. For no particularly good reason, there is a convention that things that are spinning anticlockwise are moving 'forward' (that is, have positive angular velocity), and things that are spinning clockwise are moving 'backward' (have negative angular velocity).

This is a bit like deciding that electrons have negative charge, and protons have positive charge -- physics works just fine if we swap those two, but everyone would get confused because they're used to the conventional way of doing things.

So, really, the right hand rule is essentially there so that other people (or you yourself) have an easier time making sense of what you're doing.

This may seem a bit odd, but there are many similar conventions -- for example, the convention to read English from left to right. Some languages, like Arabic, are read right to left, others like Japanese are read from top to bottom, and some texts are written with lines in alternating directions http://en.wikipedia.org/wiki/Boustrophedon.