Applications of Lines and Planes

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Anything You Want to Talk About?

Making appointments through Google calendar? In summary, the process is…

1. Go to Google calendar and sign in with your Geneseo ID (e.g., click on the link from my.geneseo, or point a browser at calendar.google.com)
2. Pull up my calendar by typing my email address in the “Search for People” box
3. Click in the calendar display where you want your appointment, as long as that spot is white and between 8:00 and 5:00, Monday through Friday
4. Fill out the resulting form with information about your appointment
5. Click “Save”.

Planes and Lines

See if you can figure out the mathematics of doing the following. For each method you figure out, also see if you can come up with an example that illustrates it. You can assume that…

• Planes are represented by scalar equations of the form Ax + By + Cz = D
• Line segments are represented by their endpoints (x₁,y₁,z₁) and (x₂,y₂,z₂)
• Points are represented by their x, y, and z coordinates.

Determining whether a point lies in a plane

Plug the point into the plane’s scalar equation. If the result is not 0 then the point isn’t on the plane, if the result is 0 the point is on the plane.

For example ???

Determining whether a line segment lies in a plane

Use the above to see if both endpoints lie in the plane.

For example ???

Determining where a line (not necessarily just a line segment, but identified by one that it contains) intersects a plane

???

Determining whether a line segment intersects a plane at all, and if so where.

???

Determining whether 2 points lie on the same side of a plane

We looked at this problem at length, since some people had started thinking about.

What I think was most helpful about the discussion was what it suggested about how to go about solving an open-ended math problem (and a lot of what we talked about isn’t just for math either):

• Talk to other people! Even if the conversation starts with “I have no f@%$-ing idea how to do this,” once you start talking more ideas will probably come out.
• Make up examples of the problem.
• Draw pictures of the examples or of the problem in general
• Write down things you know about the elements of the problem. This isn’t “write down what will solve the problem,” it’s writing down the “tools” you can think of to work with, so even writing down obvious things is helpful.

Doing some of these things with the same-side-of-plane problem, we came up with the very simple example of some points on different sides of the xy-plane (aka the z = 0 plane):

Notice that the points on one side of the plane produce positive values when plugging their coordinates into the Ax+By+Cz+D = 0 equation, while the point on the other side produces a negative value. This observation fits together with the idea that points in the place produce 0 to suggest that maybe planes split space into a positive side, a negative side, and the plane itself.

There are two choices now for pursuing this idea: look at more complicated examples to see if it holds up and if the new examples suggest ways of proving it, or go straight to proving it. Since we didn’t see a way to prove it yet, we went for another example.

This time we used the same points, but had the plane include the y axis and make a 45 degree angle to the x, passing through the (1,1,1) point. From the geometry of the situation, we could tell that point P(1,2,3) is on one side of that plane, and R(1,2,-3) is on the other. We had to come up with an equation for the plane, but to do that we could go back to ideas from the original discussion of planes, namely that vector ⟨A,B,C⟩ from Ax + By + Cz + D = 0 is normal to the plane. A plane that makes a 45 degree angle with the x axis will have a normal that also makes a 45 degree angle with the axis, pointing generally in the negative x and positive z directions:

Sure enough, when we plugged the coordinates of our 2 points into this plane’s equation, one point produced a positive result and the other a negative result. And furthermore, the point with the positive result is on the side the normal points towards.

(We didn’t have time to prove that this is a general rule, but that observation about the normal suggests the following argument: Remember the vector equation for a plane, n • PQ = 0, where n is the normal to the plane and PQ is a vector between 2 points in it. Suppose that instead of being in the plane, point Q is on the side of the plane that the normal points toward. Then n • PQ = |n| |PQ| cos a (where a is the angle between n and PQ) is positive, because a is less than π/2 and so has a positive cosine. Similarly if Q is on the side of plane opposite the normal, cos a < 0 and so the dot product is negative.)

Determining whether a point lies inside a convex polyhedron (hint: assume the planes containing the faces of the polyhedron all have their normals pointing out of the polyhedron)

???

Next

When we talked about lines, we implicitly saw functions (e.g., the r(t) functions that described lines) that produce points in space as their results. It’s time to look more deeply at such functions, and the related calculus (e.g., their limits, derivatives, and integrals).

To get started, please read “Definition of a Vector-Valued Function” and “Graphing Vector-Valued Functions” in section 2.1 of the textbook for tomorrow.

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