Question

Determine whether the given line and the given plane are parallel :

a.) x = 1 + t, y=-1, z=-2t and x = 2y +3z - 9 =0,

b.) <0, 1, 2> +t <3,2,-1> and 4x - 2z +1 = 0

Answer 1

$(\text a)\\(\text a) x=1+t,\ y=-1-t,\ z=-2t, \\\vec{a}=(1,-1,-2)\\ (\alpha) x+2y+3z-9=0, \vec{n}=(1,2,3)\\ \vec{n}\cdot\vec{a}=1\cdot1+(-1)\cdot2+(-2)\cdot3=-7\neq 0$

Hence, we can say that the given line and the given plane are not parallel.

$(\text b)\\(a) <0,1,2>\ +t\ <3,2,-1>, \\\vec{a}=(3,2,-1)\\ (\alpha) 4x-y+2z+1=0, \\\vec{n}=(4,-1,2)\\ \vec n\cdot\vec a=3\cdot4+2\cdot(-1)+(-1)\cdot 2=8\neq0$

Hence, we can say that the given line and the given plane are not parallel.