Why we only need to consider one component of the torque vector when we consider only cases in which the rotational axis is fixed in direction?

Question

In my textbook, it is said that "we will consider only cases in which the rotational axis is fixed in direction. As a result, it will be necessary to consider only one component of the torque vector."

I'm confused as to why we only need to consider one component of the torque vector when we consider only cases in which the rotational axis is fixed in direction? Also, what does it even mean by considering one component of the torque vector? Is it saying that we only consider one of the vectors that make up a torque vector (i.e. either $\vec{r}$ or $\vec{F}$)?

Answer 1

Torque is always given by:- $$\vec{\tau}=\vec{r}\times\vec{F}$$

Since, torque is a vector quantity, it can be broken down into mutually perpendicular components which add up vectorially to give the torque vector just like in the image below.

Though the example is given with acceleration $\vec a$, it is true for torque $\vec \tau$ as well. This is what is meant with components in the question. It does not mean either $\vec{r}$ or $\vec{F}$.

By one component here, they are probably trying to mean the component of the torque vector along the fixed axis because the other components of the torque vector (which are not along the axis) are automatically nullified as no rotation is allowed about any other axis (due to fixed axis).

Hope it helps.