This is a webgem on dealing with wrenches. A wrench is a force $F$ with a coaxial couple $hF.$ The word 'coaxial' means the couple is in the same direction as the force. The scalar $h$ is the pitch of the wrench, which has dimensions of length. The force acts along some line of action. Say you have a position vector $r_{?}$ to the line of action of the force. The moment of the wrench is written:

$M_{{?}_{?}}(o)=r_{}{F}_{+}h{F}_{.}$

When the pitch is zero, then the wrench is a pure force. Often, the wrench is written in the combined force$/$moment format:

On the righthand side, the first term is the pure force acting along its line of action $($i$.$e$.$ set by $r_{?})$ and the second term is the coaxial couple. We will make use of this format. First with $h=0$ and then later with $h0.$

The wrench can also be written a second way:

In the second version, on the righthand side, the first term is the pure force acting through the origin $($no moment vector$)$ and the second term is couple $($no force vector$).$ We will make use of this format too. First with $F_{*}{M}_{{?}_{?}}(o)=0$ and then later with $F_{*}{M}_{{?}_{?}}(o)0.$

For more information, see Here.

Pure Force

Consider you are given the following pure force:

First prove it is a pure force. Enter the result of this dot product.

$F_{*}{M}_{{?}_{?}}(o),(N^{2}m{)}^{}$

Next, consider an $r_{?}={}_{+}y{j}_{+}z{k}_{?}$ such that $M_{{?}_{?}}(o)=r_{}{F}_{?}.$ Assume $x=-1\mathrm{.}000000m.$ What are $y$ and $z?$

$\backslash$table$[[y_{?},\frac{?}{\text{b}\text{a}\text{r}}((m))$