Question

(1.1) Determine whether the given line and the given plane are parallel:

(a) x = 1 + t, y = −1 − t, z = −2t and x + 2y + 3z − 9 = 0,

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

Answer 1

(a)

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

The given line and the given plane are not parallel.

(b)

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

The given line and the given plane are not parallel.