# Meaning of the direction of the cross product

I was doing calculations with torque and then I came across something very confusing:

I understand that the magnitude of the torque is given by product of the displacement(from the center of rotation) at a point and the force applied at that point. This magnitude perhaps, says something about how "powerful" the rotation is. The question that confuses me, however, is what does the direction of the cross product tell? The direction is perpendicular both to the force and the displacement. What really is this the direction of?

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Cross products are indeed not intuitive. They should be replaced by the concept of a bivector, which reduces to the cross product in three dimensions but unlike the cross product, extends to other dimensions. Bivectors have an orientation in the plane in which they live. – user11266 Nov 12 '12 at 5:49

When studying angular things - torque, angular velocity, angular momentum, etc. - physicists do a clever thing to avoid having to describe curves. You see, you might be tempted to draw a curved arrow for a torque, indicating that you are twisting something around in a circular-ish way. But then when you try to add two such arrows together, all of a sudden you realize your notation no longer has a natural, intuitive meaning.

Instead, we draw the arrow pointing perpendicular to the plain of the curve you are tempted to draw. More precisely in the case of torque, perpendicular to the plain defined by the radial vector and the force vector. Note that this uniquely defines what plane your curved arrow must reside in, and, given the right-hand rule, clears up the ambiguity as to which way your curved arrow should point (if your right-hand fingers curl in the direction of the curved arrow you want to draw, your thumb points in the direction of the straight arrow you should draw instead).

It is then a simple matter to encode the magnitude of the torque/angular velocity/whatever in the length of this vector. The benefit is that you end up with straight arrows describing everything, and they add exactly as your torques should add - you have a genuine vector space, and are free to abstract away from all diagrams. And it is not even terribly counterintuitive - the torque vector is parallel to the axis around which you are applying torque. If you think about it long enough, you should be able to convince yourself that if you had to choose a single direction to define things, this is the least ambiguous.

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Your statement that adding two circular-ish arrows no longer has an intuitive meaning isn't correct. Such notation is precisely what is used for bivectors and gives them an "orientation" moreso than a direction. Orientations add algebraically just like directions do. – user11266 Nov 12 '12 at 5:46

Exactly like a force vector tells you the rate of change of momentum (both magnitude and direction), torque tells you the rate of change of angular momentum, both magnitude and direction.

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I don't really see how this makes any physical sense. An angular momentum's change in "direction" (insofar as it has such a thing), could not possibly be normal to the direction of the force applied. – McGarnagle Nov 12 '12 at 4:41
@dbaseman ... and yet it is. Check out precession of a spinning top or gyroscope on wikipedia or elsewhere. – Art Brown Nov 12 '12 at 5:02
@dbaseman There are two ways to change a vector: changing its magnitude and changing its direction. The former requires a change parallel or antiparallel to the vector. The latter requires a change perpendicular to the vector. Any arbitrary change can be decomposed into some combination of these two. – user11266 Nov 12 '12 at 6:03

It tells you whether the torque is clockwise or counterclockwise, in accordance with the right hand rule--point your thumb in the direction of the torque vector, and your fingers will curl around in the direction of rotation.

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When you apply a force to a rigid body your force might contain a component that seeks to deform the body (and this is opposed by a force equal to that component), but the remainder of the force, the part orthogonal can rotate the body. The two forces, the applied external force and the constraint force naturally lie in a plane. If you never plan to go beyond 3D and don't like to visualize the plane in which things happen, then instead you can make a convention to represent planes with a vector orthogonal to them, and establish another convention about how the orientation of the vector orthogonal represents the two different directions you can rotate in a plane. Then whenever you show someone a vector your audience/customers/etc. can wonder if you meant to refer to a plane or to a vector.

In 4D this artificial crutch is not possible, and no convention can save you, well I shouldn't say never maybe some people would sooner use complex vectors to represent the 6 independent planes that exist in 4D rather than to call a plane a plane. Of course, an imaginary vector for a plane is almost good enough, but so close to doing it properly that you might as well introduce planes as objects and be done with it.

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