Line Parallel to the Plane Equation (Final Exam Review)

NastyAccident

1. The problem statement, all variables and given/known data
Explain why the line <x,y,z> = <3,1,4> + t<4,-5,2> is parallel to the plane with equation 2x + 2y +z = 7


2. Relevant equations
The normal vector of <x,y,z> [4,-5,2] and the plane equation 2x + 2y + z = 7


3. The attempt at a solution
Well, I'm trying to review for the final exam and I'm missing a crucial notes sheet.

So, I attempted to do the dot product of the normal vector and the plane equation vector which is:

4*2 + -5*2 + 1*2 = 0

However, that didn't add up to 7 which would mean == lines.

Though, I think by writing out the dot product I technically proved perpendicularity since plane equations are based off a vector and a point. Thus, making it perpendicular to that point.

So if two bits are perpendicular to the same point then they are parallel to each other.

Any help, would be much appreciated.

konthelion

Yes, that's correct. If the line <x,y,z> = <3,1,4> + t<4,-5,2> is parallel to the plane, then its direction vector i.e. (4,-5,2) is perpendicular to the plane's normal vector.

NastyAccident

Thank You So Much!