$[$Sec$.10\mathrm{.}3]$ Wind is blowing with a speed of $20\sqrt[\phantom{2}]{10}$ in the direction of the vector $($:$0,1,3$:$).$ An airplane takes off from the origin along a straight line in the $yz-$plane with a slope of $\frac{4}{3}.$

$($a$)$ Find the velocity vector of the plane's tailwind $($the component of the wind in the direction of travel of the plane$).$

$($b$)$ Find the velocity vector of the plane's crosswind $($the component of the wind perpendicular to the travel of the plane$).$

$($c$)$ What is the speed of the plane's crosswind?