# Problem: Determine the couple moment. Use a vector analysis and express the result as a Cartesian vector.

###### FREE Expert Solution

The moment about O due to applied force at B is:

$\overline{){{\mathbf{M}}}_{{\mathbf{O}}}{\mathbf{=}}{{\mathbf{r}}}_{\mathbf{O}\mathbf{B}}{\mathbf{×}}{\mathbf{F}}}$, where rOB is the position vector between B and O and F is the resultant force.

Cross product:

The resultant force is calculated by Certasian vector notation.

We resolve each force into x- and y-components the add the repetitive components by scalar algebra.

Resultant force:

$\overline{)\begin{array}{rcl}{\mathbf{F}}_{\mathbf{n}\mathbf{e}\mathbf{t}}& {\mathbf{=}}& \mathbf{\left(}{\mathbf{F}}_{\mathbf{1}\mathbf{x}}\mathbf{i}\mathbf{+}{\mathbf{F}}_{\mathbf{1}\mathbf{y}}\mathbf{j}\mathbf{\right)}\mathbf{+}\mathbf{\left(}{\mathbf{F}}_{\mathbf{2}\mathbf{x}}\mathbf{i}\mathbf{+}{\mathbf{F}}_{\mathbf{2}\mathbf{y}}\mathbf{j}\mathbf{\right)}\\ & {\mathbf{=}}& \mathbf{\left(}{\mathbf{F}}_{\mathbf{1}\mathbf{x}}\mathbf{+}{\mathbf{F}}_{\mathbf{2}\mathbf{x}}\mathbf{\right)}\mathbf{i}\mathbf{+}\mathbf{\left(}{\mathbf{F}}_{\mathbf{1}\mathbf{y}}\mathbf{+}{\mathbf{F}}_{\mathbf{2}\mathbf{y}}\mathbf{\right)}\mathbf{j}\\ & {\mathbf{=}}& \mathbf{\sum }{\mathbf{F}}_{\mathbf{x}}\mathbf{i}\mathbf{+}\mathbf{\sum }{\mathbf{F}}_{\mathbf{y}}\mathbf{j}\end{array}}$

The position vector between two points is given by:

$\overline{)\begin{array}{rcl}{\mathbf{r}}_{\mathbf{A}\mathbf{B}}& {\mathbf{=}}& \mathbf{\left(}{\mathbf{x}}_{\mathbf{B}}\mathbf{-}{\mathbf{x}}_{\mathbf{A}}\mathbf{\right)}\mathbf{i}\mathbf{+}\mathbf{\left(}{\mathbf{y}}_{\mathbf{B}}\mathbf{-}{\mathbf{y}}_{\mathbf{A}}\mathbf{\right)}\mathbf{j}\mathbf{+}\mathbf{\left(}{\mathbf{z}}_{\mathbf{B}}\mathbf{-}{\mathbf{z}}_{\mathbf{A}}\mathbf{\right)}\mathbf{k}\end{array}}$

We also need to know some general rules for vector cross products.

$\overline{)\begin{array}{rcl}\mathbf{i}\mathbf{×}\mathbf{i}& {\mathbf{=}}& {\mathbf{0}}\\ \mathbf{i}\mathbf{×}\mathbf{j}& {\mathbf{=}}& {\mathbf{k}}\\ \mathbf{i}\mathbf{×}\mathbf{k}& {\mathbf{=}}& \mathbf{-}\mathbf{j}\\ \mathbf{j}\mathbf{×}\mathbf{j}& {\mathbf{=}}& {\mathbf{0}}\\ \mathbf{j}\mathbf{×}\mathbf{i}& {\mathbf{=}}& \mathbf{-}\mathbf{k}\\ \mathbf{j}\mathbf{×}\mathbf{k}& {\mathbf{=}}& {\mathbf{i}}\\ \mathbf{k}\mathbf{×}\mathbf{k}& {\mathbf{=}}& {\mathbf{0}}\\ \mathbf{k}\mathbf{×}\mathbf{i}& {\mathbf{=}}& {\mathbf{j}}\\ \mathbf{k}\mathbf{×}\mathbf{j}& {\mathbf{=}}& \mathbf{-}\mathbf{i}\end{array}}$

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###### Problem Details

Determine the couple moment. Use a vector analysis and express the result as a Cartesian vector.