We defined the torque exerted by force $\vec{F}$ to be $\tau =rF\text{sin}\varphi$. The quantity F is the magnitude of the force vector $\vec{F}$, and the distance r is really the magnitude of the position vector $\vec{r}$. Hence torque looks very much like a product of vectors $\vec{r}$ and $\vec{F}$. Previously, in conjunction with the definition of work, we introduced the dot product of two vectors: $\vec{A}\cdot \vec{B}=AB\text{cos }\alpha$, where $\alpha$ is the angle between the vectors. $\tau =$ $rF\text{sin}\varphi$ is a different way of multiplying vectors that depends on the sine of the angle between them.

FIGURE 12.45 shows two vectors, $\vec{A}$ and $\vec{B}$, with angle $\alpha$ between them. We define the cross product of $\vec{A}$ and $\vec{B}$ as the vector

The symbol $\times$ between the vectors is required to indicate a cross product. The cross product is also called the vector product because the result is a vector.

FIGURE 12.45 The cross product $\vec{A}\times \vec{B}$ is a vector perpendicular to the plane of vectors $\vec{A}$ and $\vec{B}$.

The right-hand rule, which specifies the direction of $\vec{A}\times \vec{B}$, can be stated in three different but equivalent ways:

These methods are easier to demonstrate than to describe in words! Your instructor will show you how they work. Some individuals find one method of thinking about the direction of the cross product easier than the others, but they all work, and you’ll soon find the method that works best for you.

Referring back to Figure 12.45, you should use the right-hand rule to convince yourself that the cross product $\vec{A}\times \vec{B}$ is a vector that points upward, perpendicular to the plane of $\vec{A}$ and $\vec{B}$. FIGURE 12.46 shows that the cross product, like the dot product, depends on the angle between the two vectors. Notice the two special cases: $\vec{A}\times \vec{B}=\vec{0}$ when $\alpha =0\text{°}$ (parallel vectors) and $\vec{A}\times \vec{B}$ has its maximum magnitude AB when $\alpha =90\text{°}$ (perpendicular vectors).

FIGURE 12.46 The magnitude of the cross-product vector increases from 0 to AB as $\alpha$ increases from $0\text{°}$ to $90\text{°}$.

EXAMPLE 12.16 Calculating a cross product

FIGURE 12.47 shows vectors $\vec{C}$ and $\vec{D}$ in the plane of the page. What is the cross product $\vec{E}=\vec{C}\times \vec{D}?$

FIGURE 12.47 Vectors $\vec{C}$ and $\vec{D}$.

FIGURE 12.47 Full Alternative Text

SOLVE The angle between the two vectors is $\alpha =110\text{°}$. Consequently, the magnitude of the cross product is

$$E=CD\text{ sin }\alpha =(2\text{ m})\left(1\text{ m}\right)\text{ sin}\left(110\text{°}\right)=1.88{\text{ m}}^2$$

The direction of $\vec{E}$ is given by the right-hand rule. To curl your right fingers from $\vec{C}$ to $\vec{D}$, you have to point your thumb into the page. Alternatively, if you turned a screwdriver from $\vec{C}$ to $\vec{D}$ you would be driving a screw into the page. Thus

$$\vec{E}=(1.88{\text{ m}}^2\text{, into page})$$

REVIEW Notice that $\vec{E}$ has units of ${\text{m}}^2$.

The cross product has three important properties:

1. The product $\vec{A}\times \vec{B}$ is not equal to the product $\vec{B}\times \vec{A}$. That is, the cross product does not obey the commutative rule $ab=ba$ that you know from arithmetic. In fact, you can see from the right-hand rule that the product $\vec{B}\times \vec{A}$ points in exactly the opposite direction from $\vec{A}\times \vec{B}$. Thus, as FIGURE 12.48a shows,

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

   FIGURE 12.48 Properties of the cross product.

   

   Full Alternative Text

   

   Full Alternative Text

2. In a right-handed coordinate system, which is the standard coordinate system of science and engineering, the z-axis is oriented relative to the xy-plane such that the unit vectors obey $î\times ĵ=\widehat{k}$. This is shown in FIGURE 12.48b. You can also see from this figure that $ĵ\times \widehat{k}=î$ and $\widehat{k}\times î=ĵ$.

3. The derivative of a cross product is

   $${\displaystyle \frac{d}{dt}}(\vec{A}\times \vec{B})={\displaystyle \frac{d\vec{A}}{dt}}\times \vec{B}+\vec{A}\times {\displaystyle \frac{d\vec{B}}{dt}}$$ (12.47)