Question $11$: $8$ Marks

$(11\mathrm{.}1)$ Let $P_0=(1,-2,3),$ and let vec$(n)=($:$3,1,-1$:$)be$ the normal vector $to$ a plane $U.$

$(a)$ Find the vector form $of$ the equation $of$ the plane $U$ passing through $P_0$ with normal

vector vec$(n).$

$(b)$ Find the shortest distance from the point $Q=(4,0,-1)to$ the plane $U,$ and give the

coordinates $of$ the foot $of$ the perpendicular from $Qto$ the plane.

$(11\mathrm{.}2)$ Let $L_{1}be$ the line given $by$:

$x=-1+3t,y=5+3t,z=2+t$

and let $L_{2}be$ the line $of$ intersection $of$ the planes:

$x-2(y-1)+3z=-1,$ and $,y-2x-1=0$

$(a)$ Find a parametric form $of{L}_2.$

$(b)$ Find the equation $of$ the plane that contains line $L_1$ and $is$ parallel $to$ line $L_2.$