Vector Cross Product/Examples/Couple Exerted by Force

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Example of Vector Cross Product

Let $\mathbf F$ be a force acting at a point $P$ on a body $B$ axially about an axis of rotation $R$ such that the distance from $P$ to $R$ is represented by the displacement vector $\mathbf d$.

Then the couple exerted on $B$ by $\mathbf F$ is defined as:

$\mathbf T = \mathbf F \times \mathbf d = \norm {\mathbf F} \norm {\mathbf d} \mathbf {\hat t} \sin \theta$


$\times$ denotes vector cross product
$\mathbf {\hat t}$ denotes the unit vector perpendicular to both $\mathbf F$ and $\mathbf d$ according to the right-hand rule
$\theta$ is the angle between the directions of $\mathbf F$ and $\mathbf d$.