Answer to Question #213616 in Linear Algebra for Tshego

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

$\implies$ We know that line parallel to plane is perpendicular to the normal vector of plane.

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

direction ratios of line is < 1, 0, -2 > and normal vector to the plane is < 1, 2, 3 >

dot product of direction ratios of line and normal is given by < 1, 0, -2 > • < 1, 2, 3 > = 1 - 0 - 6 = -5 $\neq$ 0

Hence, Given line and plane are not parallel.

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

direction ratios of line is < 3, 2, -1 > and normal vector to the plane is < 4, -2, 1 >

dot product of direction ratios of line and normal is given by < 3, 2, -1 > • < 4, -2, 1 > = 12 - 4 - 1 = 7 $\neq$ 0

Hence, Given line and plane are not parallel.