A modern private space company is currently developing a large reusable rocket for

orbital and planetary travel. The $50m$ long, $9m$ diameter, $180000kg($dry weight$),$

cylindrical rocket is being designed to re$-$enter the atmosphere of Earth or Mars at

supersonic speeds and reignite its rocket motors close to the ground for final

deceleration and vertical landing. The rocket motors gimbal $($change direction$)$ during

operation and apply a torque to the rocket to ensure the rocket remains vertical during

both ascent and landing. The thrust of the rocket motors can be varied from $880-$

$2,210kN.$

Assume the pivot center is the coordinate system origin and middle of the rocket, and

the gimballing rocket motors are attached to the bottom of the rocket. Calculate the

torque on the rocket if the rocket's orientation is given by a unit vector hat$(r)=0\mathrm{.}13$hat$()+$

$0\mathrm{.}26$hat$()+0\mathrm{.}96$hat$(k)$ and the rocket engine is applying $1000kN$ of thrust at orientation

given by the unit vector hat$(t)=-0\mathrm{.}32$hat$()+0\mathrm{.}49$hat$()+0\mathrm{.}81k.$ Draw a diagram of this

scenario and show all working.