Does the line [x,y,z] = [-2,6,5] + t[3,2,-1] lie in the plane 3x - 4y + z + 25 = 0 ? Please show work.

Answer 1

If the line
L: [x,y,z] = [-2,6,5] + t[3,2,-1]
lies in the plane
P: 3x - 4y + z + 25 = 0
Then both of the following must be true
1. [-2,6,5] must lie in the plane.
Check: 3(-2)-4(6)+5+25=0
so [-2,6,5] lies in the plane
2. the vector <3,2,-1> must be orthogonal to the vector <3,-4,1> of the plane.
Check: <3,-4,1>.<3,2,-1>=9-8-1=0
So the line is perpendicular to the normal of the plane.
Therefore the line lies in the plane.

Alternatively, locate any two points on the line and show that they both lie in the plane.

Answer 2

thank you!

Answer 3

You're welcome!