Consider the solid block and coordinate system below. Two forces $\backslash (\backslash$boldsymbol$\{$F$\}\_\{\backslash$boldsymbol$\{$A$\}\}\backslash )$ and $\backslash (\backslash$boldsymbol$\{$F$\}\_\{\backslash$boldsymbol$\{$B$\}\}\backslash )$ act on the corners of the block as indicated below.

Note that $\backslash (\backslash$hat$\{\backslash$imath$\}\backslash )$ is the unit vector aligned with the $\backslash ($ x $\backslash )-$direction, $\backslash (\backslash$hat$\{\backslash$jmath$\}\backslash )$ is the unit vector aligned with the $\backslash ($ y $\backslash )$ direction, and $\backslash (\backslash$hat$\{$k$\}\backslash )$ is the unit vector aligned with the $\backslash ($ z $\backslash )-$direction.

$($a$)$ If you represent the two forces by an equivalent system of resultant force $\backslash (\backslash$boldsymbol$\{$F$\}\_\{\backslash$boldsymbol$\{$R$\}\}\backslash )$ located at the origin $\backslash ((0,0,0)\backslash )$ and a couple $\backslash (\backslash$boldsymbol$\{$M$\}\backslash ),$ what is $\backslash (\backslash$boldsymbol$\{$F$\}\_\{\backslash$boldsymbol$\{$R$\}\}\backslash )?$

$($b$)$ If you represent the two forces by an equivalent system of resultant force $\backslash (\backslash$boldsymbol$\{$F$\}\_\{\backslash$boldsymbol$\{$R$\}\}\backslash )$ located at the origin $\backslash ((0,0,0)\backslash )$ and a couple $\backslash (\backslash$boldsymbol$\{$M$\}\backslash ),$ what is the z$-$component of the vector $\backslash (\backslash$boldsymbol$\{$M$\}\backslash )?$ You do not need to solve for the other two vector components.

$($c$)$ If you needed to find the component of $\backslash (\backslash$boldsymbol$\{$M$\}\backslash )$ that is parallel to $\backslash (\backslash$boldsymbol$\{$F$\}\_\{\backslash$boldsymbol$\{$R$\}\}\backslash ),$ would you use the dot product or the cross product between $\backslash (\backslash$boldsymbol$\{$M$\}\backslash )$ and the unit vector associated with $\backslash (\backslash$boldsymbol$\{$F$\}\_\{$R$\}\backslash )?$ Circle Your answer below. You do not need to do any calculations for this part nor provide any explanation.

Cross product

Dot product

$($d$)\backslash (\backslash$boldsymbol$\{$M$\}\_\{\backslash$boldsymbol$\{$p$\}\}\backslash )$ is the component of the moment vector $\backslash (\backslash$boldsymbol$\{$M$\}\backslash )$ that is parallel to $\backslash (\backslash$boldsymbol$\{$F$\}\_\{\backslash$boldsymbol$\{$R$\}\}\backslash$cdot $\backslash$boldsymbol$\{$M$\}\_\{\backslash$boldsymbol$\{$n$\}\}\backslash )$ is the component of $\backslash (\backslash$boldsymbol$\{$M$\}\backslash )$ that is normal $($i$.$e$.,$ perpendicular$)$ to $\backslash (\backslash$boldsymbol$\{$F$\}\_\{\backslash$boldsymbol$\{$R$\}\}\backslash ).$ Assume that you have calculated the moment vector $\backslash (\backslash$boldsymbol$\{$M$\}\_\{\backslash$boldsymbol$\{$p$\}\}\backslash ),$ but do not actually do this calculation. Write $\backslash (\backslash$boldsymbol$\{$M$\}\_\{\backslash$boldsymbol$\{$n$\}\}\backslash )$ as a function of $\backslash (\backslash$boldsymbol$\{$M$\}\backslash )$ and $\backslash (\backslash$boldsymbol$\{$M$\}\_\{\backslash$boldsymbol$\{$p$\}\}\backslash ).$