Answer to Question #212264 in Analytic Geometry for Moe

Question

Which of the following planes are parallel to the line with equation $r = (3,0,1) + \lambda(3,2,-1)$

1. $2x - 3y = 6$
2. $x - y+z=6$
3. $x - 27 + z = 6$
4. $3 y + z = 6$

Answer 1

A line and plane are parallell if the dot product of direction vector of the line and normal vector of the plane is zero.

Direction vector of given line is <3,2,-1>

Option1: normal vector is <2,-3,0>

<3,2,-1>.<2,-3,0>=6-6+0=0

Option2: normal vector is <1,-1,1>

<3,2,-1>.<1,-1,1>=3-2-1=0

Option2: normal vector is <1,0,1>

<3,2,-1>.<1,0,1>=3-1=2

Option2: normal vector is <0,3,1>

<3,2,-1>.<0,3,1>=6-1=5

Hence, options 1 and 2 are correct.