Torque and Angular Momentum as Cross Products

The Cross Product

  

Figure 2: Cross Product

$$\vec{C}=\vec{A}\times\vec{B}$$

of two vectors $\vec{A}$ and $\vec{B}$ is defined as the vector with magnitude:

$$C = AB \sin(\theta)$$

where $\theta$ is the angle subtended by the two vectors. The direction of the cross product $\vec{C}$ is always perpendicular to both $\vec{A}$ and $\vec{B}$ with the direction of the arrow determined by the right hand rule: point the fingers of your right hand from $\vec{A}$ to $\vec{B}$, and your thumb points along the direction of the cross product $\vec{A}\times\vec{B}$. NOTE:

• The magnitude of the cross product has a geometrical interpretation as the area of the parallelogram subtended by the vectors $\vec{A}$ and $\vec{B}$.
• $\vec{B}\times\vec{A}= - \vec{A}\times \vec{B}$ (i.e. they have the same magnitude but point in opposite directions.
• If $\vec{A}$ and $\vec{B}$ are parallel, $\theta=0$ and the cross product is zero.

Torque about an axis O on a point mass due to a force $\vec{F}$:

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

where $\vec{r}$ is the position vector of the mass relative to the axis of rotation O.

Angular Momentum about an axis O of a point ass m moving with velocity $\vec{v}$:

$$\vec{L}= \vec{r}\times\vec{p}$$

where $\vec{r}$ is the position of the mass relative to the axis and $\vec{p}=m\vec{v}$ is its linear momentum.