Let L be the line containing the point P215 and perpendicula

Let L be the line containing the point P(2,1,5) and perpendicular to the plane 4x12x2x3=2.

Find a vector equation for L.

L:x(t)= ??

At what point does L intersect the yz-plane?

Intersection: ??

Solution

L is the line containing the point P(2,1,5) and perpendicular to the plane 4x₁2x₂x₃=2

normal of the plane is direction vector of the line

normal of the plane=<-4,-2,-1>

so direction vector of the line n=<-4,-2,-1>

L is the line containing the point P(2,1,5)

vector equation of line is L = P+ tn

L(x,y,z)= (2,1,5) + t<-4,-2,-1>

L(x,y,z) =<2-4t,1-2t ,5-t >

when L intersect the yz-plane, x =0

2-4t =0

t =-1/2

point of intersection =(2-4(-1/2),1-2(-1/2) ,5-(-1/2) )

point of intersection =(0,0,11/2 )

L intersect the yz-plane at (x,y,z)=(0,0,11/2 )