Parallel plane containing a line

In summary, the task is to find the plane that contains the line x=3+2T, Y=t, Z=8-t and is parallel to the plane 2x + 4y +8z =17. It can be done by converting the line equation into a parametric form and using a point on the line to determine the value of D in the plane equation. This will make the plane parallel to the given plane.
  • #1
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Homework Statement
Find the plane that contains the line x=3+2T, Y=t, Z=8-t and is parallel to the plane 2x + 4y +8z =17.

The attempt at a solution

It seemed to me that since the plane is parallel to the plane 2x+4y+8z=17, it must be of the form 2x+4y+8z+D=0 so that it would be parallel.
(Is that true?)

Anyways, I then converted the equation of the line out of parametric into (x-3)/2 = Y = -(Z-8).
(I know that part's right.)

But now I'm not sure what to do. Ideas?
 
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  • #2
Yes, your parallel plane equation is correct. Now just substitute x, y and z from the line equation into the plane equation and figure out what D must be.
 
  • #3
It is, in fact, sufficient to use a single point on the line. Do you see why? For example, it should be easy to find D such that (3, 0, 8) (taking T= 0 in the equations of the line) is on the plane.
 

What is a parallel plane containing a line?

A parallel plane containing a line is a plane that does not intersect with the given line, but maintains a constant distance from it at all points. This means that the line and the plane do not share any common points, but are always the same distance apart.

How can you determine if a plane is parallel to a given line?

A plane is parallel to a given line if the plane's normal vector is perpendicular to the line's direction vector. This means that the dot product of the plane's normal vector and the line's direction vector is equal to zero. If this condition is met, the plane is parallel to the line.

What is the equation of a parallel plane containing a line?

The equation of a parallel plane containing a line is in the form Ax + By + Cz + D = 0, where (A,B,C) is the normal vector of the plane and D is a constant determined by the given line's coordinates.

Can a parallel plane contain more than one line?

Yes, a parallel plane can contain an infinite number of lines. This is because any line that is parallel to the given line will also maintain a constant distance from the plane and therefore be contained within it.

What is the relationship between parallel planes and parallel lines?

If two planes are parallel, then any lines contained within those planes will also be parallel. However, if two lines are parallel, it does not necessarily mean that the planes containing those lines are parallel. The planes may be parallel or they may intersect at a point.

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