Line Parallel to the Plane Equation (Final Exam Review)

[NastyAccident]

Homework Statement

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

Homework Equations

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

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!