Now let’s return to torque. As a concrete example, FIGURE 12.49 shows a long wrench being used to loosen the nuts holding a car wheel on. We’ve established a right-handed coordinate system with its origin at the nut, so force $\vec{F}$ exerts a torque about the origin. Let’s define a torque vector

FIGURE 12.49 The torque vector.

If we place the vector tails together in order to use the right-hand rule, we see that the torque vector is perpendicular to the plane of $\vec{r}$ and $\vec{F}$. The angle between the vectors is $\varphi$, so the magnitude of the torque is $\tau =rF\left|\text{sin}\varphi \right|$.

You can see that the scalar torque $\tau =rF\text{sin}\varphi$ we’ve been using is really the component along the rotation axis—in this case ${\tau }_x$—of the vector $\vec{\tau }$. This is the basis for our earlier sign convention for $\tau$. In Figure 12.49, where the force causes a ccw rotation, the torque vector points in the positive x-direction, and thus ${\tau }_x$ is positive.

EXAMPLE 12.17 Wrench torque revisited

Example 12.8 found the torque that Luis exerts on a nut by pulling on the end of a wrench. What is the torque vector?

VISUALIZE FIGURE 12.50 shows the position vector $\vec{r}$, drawn from the pivot point to the point where the force is applied. The figure also redraws the force vector $\vec{F}$ at the pivot point, not because force is applied there but because it’s easiest to use the right-hand rule if the vectors are drawn with their tails together.

FIGURE 12.50 Calculating the torque vector.

FIGURE 12.50 Full Alternative Text

SOLVE We already know the magnitude of the torque, $17\text{N}\text{m}$, from Example 12.8. Now we need to apply the right-hand rule. If you place your right thumb along $\vec{r}$ and your index finger along $\vec{F}$, which is somewhat awkward, you’ll see that your middle finger points into the page. Alternatively, make a loose fist of your right hand, then orient your fist so that your fingers curl from $\vec{r}$ toward $\vec{F}$. Doing so requires your thumb to point into the page. Using either method, we conclude that

$$\overrightarrow{\tau }=(17\text{ N m}\text{,}\text{ into page})$$