Why is torque the cross product of the radius and force vectors?

Question

I understand the torque vector to be the cross product of the radius (moment arm) and force vectors, but that means the torque would be perpendicular to the radius and force vectors, which makes no sense to me, e.g. a force applied tangent to the surface of a car tire creates a torque along the line of the axle.

I'm pretty sure I am just misunderstanding a simple formula, so I wanted to make sure.

And, when you use the formula for torque, is torque defined as a vector or just a scalar? I would think it would be a vector.

Answer 1

We need a quantity that conveys information about the direction of the radius vector and the direction of the force vector.

Two vectors determine a plane. So we need a quantity that specifies a plane. One way to do that is to specify the vector normal to the plane. That's what the cross product does for us. There is an ambiguity as to direction: there are two normal vectors to a plane. We solve this by choosing one arbitrarily: we decide by convention to use the right-hand rule.

There are other ways to represent a torque that some would argue are more natural, for example, the bivector. These other ways are usually extensible to dimensions higher than three, whereas the cross product works only in three dimensions. Well, we live in a world having three spatial dimensions. That fact, and years and years of usage and tradition, has cemented the cross product into our toolbox.

The cross product has a few oddities associated with it, but it does the job. Some people think we should do away with the cross product. It might be nice to do that and use a more natural mathematical construct, but trying to make a change like that is like rolling a very large boulder up a hill.

Answer 2

You are right in saying that the torque points along the line of axle. This doesn't make sense intuitively, but if you look at the formalism of angular momentum vector, this would be obvious.

So angular momentum is defined as $L = r \times p$. And torque is defined as $\tau = r \times F$ . It is clear that $$\frac{d\vec{L}}{dt} = \frac{d\vec{r}}{dt} \times \vec{p} + \vec{r} \times \frac{d\vec{p}}{dt} = \frac{\vec{p}}{m} \times \vec{p} + \vec{r} \times \vec{F} = \vec{r} \times \vec{F} = \vec{\tau}$$

When you look at the car tire case. The friction creates a torque along the axle, which increases the angular momentum of tire in that axle direction. What does that mean? Since $$L=r \times p$$ and $r$ does not change, $p$ must increase. Which makes sense, since friction makes the wheel spin faster!

In summary, the direction of torque vector is determined by the definition of angular momentum and angular velocity vectors which capture more measurable kinematic quantities. This formalism is a bit counterintuitive at first. But you will eventually get used to it...

Answer 3

A force acts upon a line of action in 3D space. The force vector can be anywhere along this line and it won't change the situation.

Torque is the moment of force because it conveys the (perpendicular) distance where this force acts upon. Any component of location along the line of action needs to be ignored and this is achieved with the vector cross product.

Other common moments in mechanics are:

• Moment of Rotation - Linear velocity is the moment of rotation because the velocity of a point A depends on the perpendicular distance to the axis of rotation $$\mathbf{v}_A = \mathbf{r} \times {\boldsymbol \omega}$$ where $\mathbf{r}$ is the location of the axis relative to A.
• Moment of Force - Torque is the moment of force because the equipollent torque at a point A depends on the perpendicular distance to the line of action $${\boldsymbol \tau}_A = \mathbf{r} \times \mathbf{F}$$ where $\mathbf{r}$ is the location of the line of action relative to A.
• Moment of Momentum - Angular momentum is the moment of momentum because the angular momentum about a point A depends on the perpendicular distance to the axis of momentum $$\mathbf{L}_A = \mathbf{r} \times \mathbf{p}$$ where $\mathbf{r}$ is the location of the axis of momentum relative to A.

All of the above are similar because they are manifestations of the same law. The law described by Julius Plücker when he used the moment of a line to describe the location of a 3D line in space. Rotation, Momentum and Forces all contain lines in space. Their moments are a set of (homogeneous) coordinates describing the closest point on the line to the point of measurement.