Intersections of Line and Planes Part 1

This lesson shows how to find intersections of lines and planes in Three-Space. This is the first part of a two part lesson. This lesson was created for the Calculus and Vectors (MCV4U) course in the province of Ontario, Canada.

[AlRichards]

The plane's equation is in scalar or Cartesian form so the [2, -3, 1] vector is perpendicular to the plane. The direction vector of the line [3, 6, -4] is parallel to the line. If the dot product of the line and the normal from the plane was equal to zero, then the line would run parallel to the plane. Since the dot product of the normal and direction vector isn't 0 then the line does not run parallel to the plane, so it must intersect the plane somewhere. Hope this helps, Al.

[paulus]

Thx a lot Richard! Very clear and straightforward explaination!

[AlRichards]

@partikla86 To get a direction vector for a line either find a vector between 2 points that you know are on the line, or if you know a line parallel to it then you can use that parallel line's direction vector or any multiple of it.

[donaldmclachlan]

Hi!

It has been 30+ years since I learned 3D intersections of lines and planes. With your videos I am once again able to find the point of intersection of lines and planes.

Now, assuming someone drew a non right angle parallelogram on the plane. Is there a standard mechanism to determine whether the point of intersection is within that area of the plane?

Fyi, for self edification I've written a 2D ray tracing program that I am trying to extend to 3D. The problem I'm foreseeing is if not all of all walls are right/plumb/rectangular, how can I detect whether the ray has hit the wall, or just the plane outside beyond the area of the wall on the plane.

Any help would be greatly appreciated.

Thanks,
Don

[yhenbermudez]

How can I get the coordinates of the 2 line, if only equations are given. since coordinates are necessary to get the intersection