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Why is the cross product used in torque $$t = r\times F ~?$$

I know that the torque is the rate of change of angular momentum and angular momentum is defined as the cross product $$L = r\times P$$ but why is this appearing?

I know that the cross product is a vector of the magnitude of the determinant of the other two vectors and perpendicular. Can someone derive or explain why the cross product is there?

Qmechanic
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ramanujans alkhazarim
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2 Answers2

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I'll use $\vec{\tau}$ to denote torque, and reserve $t$ for time. As you said, torque is the rate of change of angular momentum

$$\vec{\tau} = \frac{d\vec{L}}{dt} = \frac{d}{dt}\left(\vec{r}\times \vec{p}\right)$$

from your definition for angular momentum. Then by the product rule,

$$\vec{\tau} = \left(\frac{d\vec{r}}{dt}\times \vec{p}\right) + \left(\vec{r} \times \frac{d\vec{p}}{dt}\right)$$

The linear momentum, $\vec{p}$, is in the same direction as the velocity, $\vec{v} = d\vec{r}/dt$, so the first term vanishes (the cross product of parallel vectors is zero). So we are left with

$$\vec{\tau} = \vec{r} \times \vec{F}$$

since force is the rate of change of the linear momentum.

Aiden
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The cross product is not a vector, it's an axial vector. They are geometrically different. The cross product of 2 vectors, $A_i$, $B_i$, is really:

$$ \bar{C}_{ij} = A_iB_j - A_j B_i $$

which is the antisymmetric part of a rank-2 tensor. Since that has $(3^2-3)/2 = 3$ independent components, we can construct:

$$ C_i = \frac 1 2 \epsilon_{ijk}\bar{C}_{jk} = \epsilon_{ijk}A_jB_k$$

which rotates just like vector, but is even under coordinate inversion, aka parity, (vectors are odd), hence the name: axial vector.

Since Newtons Laws are parity (and time inversion) symmetric, it restricts how they can be formulated.

(Note: the discovery of parity violation in beta decay was because the momentum of the decay electron was aligned with angular momentum of the nucleus. As you pointed out, angular momentum is the cross product 2 vectors, so it is an axial vector. That a vector $\vec p$ was proportional to an axial vector $\vec J$, was Nobel prize level shocking, at the time).

JEB
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