Re: Why is the cross product defined the way it is (Gyroscope)

The cross product is just shorthand invented for the purpose of quickly writing down the angular momentum of an object. It is not necessary for understanding the motion of the gyroscope (see http://hyperphysics.phy-astr.gsu.edu/Hbase/gyr.html for a cross product free explanation of precession), although it does make writing down the physics easier.

Here's how the cross product arises naturally from angular momentum. Recall that if we have a fixed axis and an object distance r away with velocity v with mass m that is moving around the axis in a circle, the magnitude of the angular momentum is just m|r||v|, where |r| is the magnitude of vector r. But what direction should the angular momentum vector point in? Well, if you follow the path of the object, it lies in a plane, an infinite two-dimensional surface. One way to represent a plane is to write down two different vectors that lie in the plane.

Another method used by mathematicians to represent a plane is to write down a single vector that is normal to the plane (normal is a synonym for perpendicular). If a plane is a flat sheet, the normal vector points straight up. Now, for any plane, there are two vectors that are normal to it, since if a vector n is normal to a plane, -n will be normal as well. So how do we determine whether to use n or -n? A long time ago, physicists just made an arbitrary decision known today as the right hand rule. Given vectors r and v, just curl your fingers from r to v and the thumb points in the direction of the normal used (see http://hyperphysics.phy- astr.gsu.edu/hbase/vvec.html#vvc5 for a picture) .

I want to make clear here that we could just have easily used a left-hand rule from the start. It really doesn't matter as long as we are consistent, and use the same direction for all of our calculations. Tradition (and the fact that the vast majority of humans are right handed) means that today we use the right hand rule, but the choice was arbitrary.

Ok, but what if the velocity v is not perpendicular to the distance to the axis r? In other words, what if the object isn't tracing out a perfect circle around the axis point? Well, v is just a vector, so we break it into two parts, one (v_r) that points in the same direction as r, and one (v_perp) that points in the direction perpendicular to r. Now v_perp is the part of the motion that looks exactly like the object is moving around the circle, so that is what contributes to angular momentum. The vector v_r contributes nothing to the angular momentum, but v_perp contributes m|r||v_perp|. So that's exactly how we define r x v. We use the right hand rule to find the direction, and the magnitude is exactly set to be m|r||v_perp|. Now |v_perp| can also be found as |v| sin q, where q is the angle between r and v, so sometimes you see the cross product defined this way.

The cross product is not commutative, since a x b = -b x a, but it is distributive, so a x (b + c) = a x b + a x c. This means that you can sum the effects of two angular momentum vectors into a singular angular momentum, which is one reason that angular momentum is so useful as a concept. In particular, torques (which change angular momentum) can also be written as cross products, and understanding the gyroscopes motion is all about seeing how the angular momentum (the spinning) changes under a torque (gravity, or another force).

One more note about the importance of the normal vector, so you can see a completely different use of the cross product. Normal vectors are perpendicular to everything in a certain plane. One way to see if two vectors are perpendicular is to check if they have dot product 0. If (a,b,c) is the normal vector and (x,y,z) is a point in the plane, the dot product of (a,b,c) and (x,y,z) is ax + by + cz. Therefore another way to write a plane is as the set of all points that satisfy the equation ax + by + cz = 0 (see http://mathworld.wolf ram.com/NormalVector.html for more of the mathematics of normal vectors).

Mark