Line Parallel to the Plane Equation (Final Exam Review)

In summary, the line <x,y,z> = <3,1,4> + t<4,-5,2> is parallel to the plane with equation 2x + 2y + z = 7 because its direction vector is perpendicular to the plane's normal vector. This can be shown by taking the dot product of the two vectors, which results in 0, indicating perpendicularity. This also proves that the line is parallel to the plane, since two vectors perpendicular to the same point are parallel to each other.
  • #1
NastyAccident
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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.
 
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  • #2
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.
 
  • #3
Thank You So Much!
 

1. What is the equation for a line parallel to a given plane?

The equation for a line parallel to a given plane can be expressed as Ax + By + Cz = D, where A, B, and C are the coefficients of the x, y, and z variables and D is a constant.

2. How do you determine if a line is parallel to a plane?

A line is parallel to a plane if its direction vector is perpendicular to the plane's normal vector. This can be determined by taking the cross product of the line's direction vector and the plane's normal vector. If the result is a zero vector, the line is parallel to the plane.

3. Can a line be parallel to multiple planes?

Yes, a line can be parallel to multiple planes. This occurs when the line's direction vector is perpendicular to the normal vectors of each plane.

4. How many solutions can a line parallel to a plane have?

A line parallel to a plane can have an infinite number of solutions. This is because any point on the line can satisfy the equation of the plane.

5. Is there a specific method for finding the equation of a line parallel to a plane passing through a given point?

Yes, to find the equation of a line parallel to a plane passing through a given point, you can use the point-normal form of the equation of a plane. This involves substituting the given point into the equation of the plane and solving for the constant term D. This value can then be used to write the equation of the line in the form Ax + By + Cz = D.

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