Intersections of Planes Example

A question in class today pointed out some minor detail that I left out in finding the equation of a line which is the intersection of two planes. Lets look at an example where the method in class does not work, see how to fix it, and then look at what went wrong.

Lets consider the planes $2x+y+z=2$ and $4x+2y-3z=4$. If we set $z=0$, we are left with the two equations $2x+y=2$ and $4x+2y=4$, which have infinitely many solutions. We can't continue.

A good way around this is simply to set $x=0$ or $y=0$ and solve the way we did in class. If we set $x=0$, we are given the equations $y+z=2$ and $2y-3z=4$. These have a solution of $y=2$ and $z=0$, giving us the point $(0,2,0)$. Now we are all set. We can cross product the normal vectors from each plane to get a vector pointing in the direction of the line and finish by writing the equation.

What went wrong? Geometrically, setting z=0 is equivalent to looking for the intersection of the three planes $2x+y+z=2$, $4x+2y-3z=4$, and $z=0$. In this case the line of intersection of $2x+y+z=2$ and $4x+2y-3z=4$ is contained in the plane $z=0$, so we gain no information by setting $z=0$. In the picture below $2x+y+z=2$ is the red plane, $4x+2y-3z=4$ is the blue plane, and $z=0$ is the yellow plane. Notice how the line of intersection of the red and blue planes is contained in the yellow plane.

We fixed this by setting $x=0$. Which is illustrated in the following graph(the red and blue are as above and $x=0$ is yellow.

While the intersection of the planes is not in the frame it is clear that they will intersect in a point, giving us a point which is on our line.

3D visualization

Before I get going with the topic of today's post I would like to add another resource I found for understanding the geometry involved in the dot product and cross product. Just follow the links to play.

While it is possible to solve homework questions without having a good handle on the geometry of the functions we are dealing with, I wouldn't advise it.  Here are some 3 dimensional function graphing applets that you can use.  For lines in 3 dimensions it is best to use the parametric equations, making this a good choice:

For graphing planes the following works well(make sure and use a * for multiplication, so 3x=3*x):

First Real Post: Book-keeping and Vectors.

Hopefully everyone received the email with the syllabus. In case you didn't if put it in the "Downloads" section of this blog.  I recommend that everyone in the course "follow" this blog as I will be posting helpful information including possible extra-credit opportunities.

On to the course material.  We have so far covered vectors up to the start of the cross product.  Here is a good video covering the dot product from MIT Open-Courseware.

Here is a video covering the cross-product:




Comment to this post with more online resources for the course material up to this point. This will count toward your participation grade. I'll give you until class on tuesday to comment, after which I'll post some resources I've found.