Hi everyone :)

We can determine the position of a plane and a straight line in a three-dimensional coordinate system by using the following methods. There are 3 cases that can occur and which are described below: 

The plane and the line:

illustration of the following cases                                                                                                              

1. Intersect each other

The planes have one point in common ( here "a")

the red line shows this case

Straight line: g:x = (3/4/7) + s*(2/1/-1)    plane: p = 2x1 + 5x2 - x3 = 49

It equals:

x1 = 3+ 2s

x2 = 4+ s

x3 = 7 - s

-> if you don't get where the numbers comre from, just look at the 3 of the support vector and the s*2 from the direction vector

Now you put in the equation of the line into the one of the plane:

2*(3+2s) + 5*(4+s) + (7-s) = 49

6 + 4s + 20 + 5s - 7 + s = 49

10s = 30  

If you divide it by 10:

s = 3

If you now put in 3 for the "s" of the equation of the straight line you get the point where the line goes through the plane:

(3/4/7) + 3*(2/1/-1) = a (9/7/4)  -> our intersection point.

2. Are parallel:

The plane and the line have no intersection points. You do the same as before: putting  the line equation into the one of the plane and solve the system.

g:x = (3/3/2) + s*(2/-4/4)                          p: x1 +3x2 + 2,5x3 = 0

x1 = 3 + 2s

x2 =3 - 4s

x3 = 2+ 4s

3 + 2s + 3*(3 - 4s) + 2,5*(2 + 4s) = 0

3 + 2s + 9 - 12s + 5 + 10s = 0

17 = 0 -> this is a wrong statement, so the line and the plane are parallel.

3. Are equal:

They have infinite intersection points:

the blue line shows this case

Remember our plane from the beginning? This time we use it again, but another euqation of a straigth line:

straight line: g:x = (3/8/-3) + s*(2/-1/-1)                                     plane: p = 2x1 + 5x2 - x3 = 49 

The x1, x2 and x3 can be easily read, so i leave them out and start with putting g:x int the plane:

2*(3 + 2s) + 5*(8 - s) + 3 - s = 49

6 + 4s + 40 - 5s + 3 + s = 49

0 = 0*s -> this equation has endless solutions, so g lays in p

Conclusion

As we have seen, we can mathematically prove the relative position of a plane and a straight line by using the methods from above. We can remember, if the solution from our line equation is true for every "s", the two lay in each other. If it's true for one "s", then they intersect each other and if it isn't true for any "s", they are parallel to each other.

Have a nice day :)

Source

Text

Ernst Klett Verlag, Lambacher Schweizer Seite 262 ( 1.Auflage)  (translated)

https://www.youtube.com/watch?v=hx66kNN4Cks (translated)

http://portal.tpu.ru:7777/SHARED/k/KONVAL/Sites/English_sites/G/p_PositionPL_f.htm

https://de.serlo.org/mathe/geometrie/analytische-geometrie/lagebeziehung-von-punkten-geraden-und-ebenen/lagebeziehung-einer-geraden-und-einer-ebene/lagebeziehungen-von-geraden-und-ebenen (translated)

Pictures:

https://upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Plane-line_intersection.svg/1200px-Plane-line_intersection.svg.png

https://qph.ec.quoracdn.net/main-qimg-3223c4ded2c392b7f32a44f09dcfd63c

https://assets.serlo.org/legacy/6251_XD98IJgLUP.png

https://qph.ec.quoracdn.net/main-qimg-3223c4ded2c392b7f32a44f09dcfd63c