Why are the components of position vectors along the axis of rotation not needed for determining the torque in a rigid body?

$A$ solid cone hangs from a frictionless pivot at the origin $O$,as shown. If $\hat{i}$,$\hat{j}$,and $\hat{k}$ are unit vectors,and $a, b$,and $c$ are positive constants,which of the following forces $\vec{F}$ applied to the rim of the cone at a point $P$ results in a torque $\vec{\tau}$ on the cone with a negative $z$-component $\tau_z$?

$A$ force of $(2\hat i - 4\hat j + 2\hat k) \ N$ acts at a point $(3\hat i + 2\hat j - 4\hat k) \ m$ from the origin. The magnitude of torque is ......... $N-m$.

Let a force $\vec{F} = -F\hat{k}$ act on the origin of a Cartesian frame of reference. The moment of force about a point $(1, -1)$ will be

What is the torque of a force $\vec{F} = 3\hat{i} + 7\hat{j} + 4\hat{k}$ about the origin,if the force acts on a particle whose position vector is $\vec{r} = 2\hat{i} + 2\hat{j} + 1\hat{k}$?

$A$ rod of mass $1 \ kg$ is kept on a frictionless table and is acted upon by $2$ forces as shown. We want to replace these $2$ forces by a single force so that the effect on the rod is the same. At what distance $x$ (in $m$) from the center $C$ of the rod should we apply this single force?