Planes in Space

The purpose of today's lesson is to explore planes and work up to finding a way to write their equation. We will be incorporating ideas the dot product and the cross product from previous lessons to help us out.

Teacher Note: Below is a crash course on finding the equation of a plane if you are a little rusty or have never thought about this concept before.

To get started, I want my students to think about a very simple plane. I show them my model of the 3D coordinate plane and ask students what is true about every single ordered triple on the xy-plane (the green one) and to find an equation for this plane. I give them a minute or so to discuss with their table group and then we will share out answers. The list below is what I would hope to get out of this discussion; if they don't say these things on their own I will ask questions to get these ideas out.

1. The z-coordinate is always zero for every single point on the xy-plane since none of the points are moving up or down from the green plane.
2. If the z-coordinate must be zero, then the equation of the plane is z = 0 since the x and y coordinates can be anything.

After our very simple first example, we are ready to look at a plane that is not so easy. I give students this worksheet (the answer key is here) and have them work on as much as they can independently with their table groups. This is not the easiest problem to do, but I tried to provide enough scaffolding in the questions so that students can do a lot of the work on their own without reaching a roadblock. My thought was to activate the prior knowledge of finding the angle between two vectors (question #1) and then to establish a vector on the plane and one it is orthogonal to (questions#2a and #2b). Then, students simply use the formula for the angle between two vectors and plug in the orthogonal vectors.

Inevitably, there will be questions as students work. It is tough to resist the urge to just give them the step by step process, but I try to keep them focused on the process of plugging in the two vectors we know are perpendicular into our formula.

I want to make sure that all students at least have a chance to attempt most of question #2 on the worksheet. Once that happens we will start to go through the questions. I will usually choose a student who made sense of the fact that vector PQ and vector n are perpendicular and can explain the formula we used and the algebraic simplification. I will have this student share their work on the document camera and have the class try to interpret his or her work. Many of my students have difficulty finding the component form for vector PQ, so I make sure to spend some time on that.

After question #2 we outline the process together. Here are the steps that I am looking for:

1. Find a vector normal to the plane.
2. Find a vector on the plane.
3. Plug those vectors into the formula for the angle between two vectors. Since they are perpendicular, plug in 90° for θ.
4. Simplify as much as you can.

When we get to question #3, I will ask students what steps may be problematic. They usually realize that we do not know a vector normal to the plane, so I ask them to discuss with their table groups how to do that. Once the class realizes that we can use the cross product to get an orthogonal vector, I will have them proceed to finish the problem.

Finally, question #4 asks to find the intersection of two planes. If students get stuck, I ask them to think of a way to find the intersection of two lines in two-dimensional space. This will usually signify the possibility of algebraic substitution.

At the close of the lesson, I will once again ask the class to summarize the steps of finding the equation of a plane. I will usually assign a few problems from our textbook for additional practice.