$($A$2)$ pts each$]$ Consider the plane $P_1$:$x-y+3z=7.$

a$.$ Explain using a complete sentence why any line perpendicular to the plane $P_1$ must be parallel to vector vec$(v)=($:$1,-1,3$:$).$

b$.$ Find point $Q,$ the point of intersection of the lines

$L_1$:$x=1+t,y=-5+5t,z=4+2t$ and $L_2$:$x=4+s,y=2+s,z=-2-4s.$

c$.$ Use part a to find parametric equations for the line that is perpendicular to plane $P_1$ and that passes through the point $Q$ you found in part $b.$

d$.$ Find the coordinates of the point where the line you found in part $c$ intersects the $xy-$plane.