# Understanding cross product and direction of torque

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1. May 6, 2016

### Orha

1. The problem statement, all variables and given/known data
Hi everyone,
I am a first year physics student and we recently learned about torque.
Every time I think I understand it something else comes up to confuse me - this time it is the direction. I tried looking in the forum and generally in google, but everyone only explains the right hand rule over and over again and that is not my problem. I will try to explain my difficulty as best as I can, I hope it will be clear.

I understand that cross products of 2 vectors give as a result a vector that is perpendicular to the plane of the two original vectors. I understand how to use the right hand rule and how to derive the direction of torque and I understand that mathematically it turns out to be perpendicular to the forces applied on the object.

My problem is that it doesn't make any sense logically to me that the direction of torque should be perpendicular to the plane created by the force applied to the object and the location relative to the axis it is being applied to.

2. Relevant equations
$\tau = r × F$

3. The attempt at a solution

When I asked my lecturer she told me that the direction of torque basically represents whether the object is moving clockwise or anti-clockwise. So I understood that if the force was in the direction i and located on j then the torque is directed towards ixj=k and this k does not mean "the third dimension" or "pointing in or out of the page" but rather just "turning clockwise or anticlockwise" for -k or k. She said that it turns out it means both, or maybe I misunderstood her. This confused me even more.

Torque by definition is "tendency of a force to rotate an object about an axis,[1] fulcrum, or pivot" (Wikipedia, I know, not the most trustworthy place, but good enough for my cause I think). By this definition I would expect that the direction for a torque for a given force would have to do with the direction of the force and the location relative to the axis (like it is) but remain on the plane they create.

So to conclude - why is the direction of torque perpendicular to the plane of the force and relative location from the axis that it is being applied to?
Sorry if it took me too long to explain, I hope it is clear enough, tell me if anything needs more explanation as to what my difficulty is.
Any help would be greatly appreciated.

Last edited by a moderator: May 6, 2016
2. May 6, 2016

### Orodruin

Staff Emeritus
No, the torque gives you the direction of the axis. However, this is not always true unless your inertia tensor is proportional to unity. I would start by looking at the angular momentum. Do you underdtand why it is what it is? (Of course, in the end it is a matter of definition, but you can try to build intuition for the definition.)

3. May 6, 2016

### Staff: Mentor

First, we live in a right-handed world. The notion of torque being defined via the right-handed rule is an example of the this. Its a convention that once adopted must be threaded through all of your equations and definitions of torque, angular momentum and angular velocity.

If you recall that angular momentum points along the axis of rotation and its defined as: L = r x p which by convention uses the right-handed rule

and that torque is the time rate of change of angular momentum: T = r x F then you are forced to use the same right-handed rule for torque.

Angular momentum is defined along the axis of rotation because its the one direction which is constant for a rotating body and it makes working with these problems easier in a vector sense.

In a sense. math is used to augment the reality of a physical system with concepts like vectors that help you work with and understand how the system works.

4. May 8, 2016

### Orha

First off I wanted to say thanks for the reponses.

It seems to me you say we define the torque according to the angular momentum so that explains the direction of torque (the reason this is weird for me is because in class we defined torque before we defined angular momentum and only after concluded that it happens to be the rate of change). And then to explain why the direction of angular momentum is the way it is you basically say "because it's convenient". I agree that it is great when something turns out to be convenient, but why can we define it that way?

This might be a very basic question but, I understand that in math we are allowed to define whatever we want, as long as we are consistent, but once we add meaning to it in physics it seems to me that is no longer the case.
In this example sure you can define angular momentum any way you want to so it can be convenient, but once you define torque accordingly, and the torque has a physical meaning that is "tendency of a force to rotate an object about an axis,[1] fulcrum, or pivot" then how do you know that the way you defined it in the first place actually represents that?

5. May 8, 2016

### Orodruin

Staff Emeritus
That is not the physical meaning, it is an interpretation in words of the physical meaning. The physical meaning of torque is the time derivative of the angular momentum. This is defined in a way consistent with the definition of angular momentum - this is something you have to check mathematically.

The judge for physics in the end is whether the theory describes experimental results or not. There is nothing particular for torque and angular momentum here. It is the same for force and linear momentum.