# Torque question -- why defined as r×f and not f×r?

1. Dec 2, 2014

### Vijay.V.Nenmeli

Hello everyone,
Quick question.
Could anyone explain to me why torque is defined as r×f and not f×r.Also, how do we know that taking the direction of the vector as perpendicular to the plane is valid?

2. Dec 2, 2014

### Staff: Mentor

It's a convention, just like choosing the negative direction of the x-axis to be to the left of the origin is a convention. You could do it the other way as long as you were consistent about doing it the other way everywhere - for example, you would also have to switch the order in the definition of angular momentum.

It leads to a mathematically convenient way of describing torques and angular momenta. As these have both magnitude and orientation in space, it's natural to describe them as vectors perpendicular to the plane of movement. As a historical note, torque and angular momentum were discovered before vector calculus was invented... There's a thread on this history somewhere around here.

3. Dec 2, 2014

### SteamKing

Staff Emeritus
The order here is important because the cross product does not commute; that is r x F ≠ F x r .

4. Dec 2, 2014

### Vijay.V.Nenmeli

Thanks a lot Nugatory,
But this method of taking perpendicular vectors is, as I see it, merely a way to simplify calculations. Does it affect the calculations in any way I.e If I were to take a different vector to represent torque, say one that was aligned at 45 degrees to the plane of F and R,, I'd get a different vector. Less convinient to work with, maybe, but how do we know that the 45 degree vector is not right and the perpendicular is
??

5. Dec 2, 2014

### A.T.

A convention is neither right nor wrong.

6. Dec 2, 2014

### Staff: Mentor

There are exactly two vectors (of a given magnitude) that can be perpendicular to the plane of F and R: one up and one down. There are an infinite number of vectors of that magnitude that can be at a 45 degree angle to that plane (imagine a cone with its point just touching and its axis perpendicular to the plane - all vectors along the surface of that cone are at the same angle relative to the plane). Because there are only two possible rotations in a plane, clockwise and counter-clockwise, it's easy to map the the two possible directions of a perpendicular vector to the two possible rotations.

That doesn't make a 45-degree convention "wrong" (as A.T. has pointed out above, a convention cannot be wrong), but it is a fairly strong hint that we'll get better answers faster if we use the cross-product.