Each year in the spring my pre-calc kids come to the four brief sections in our text that deal with parametric equations and vectors. (I think of this chapter as the catch-all chapter, anything we might have missed that is in the California or Texas Standards). And then I throw in about three more weeks of work about vectors that I created because I think it is a)beautiful and b) really important. And I share with my students that it seems incredible to me that they, the best and brightest mathematics students in our school, are nearing the end of their high school education (many are seniors) and they can't do one of the most simple acts of coordinate geometry; that is, "Given two points, write the equation of a line containing the two points" (How does the standard for your school read?).

They look at me in wonder, confusion, and perhaps some doubt of my sanity. After all, we have done that thousands of times. I continue to bemoan their lack of ability until finally someone will challenge me...."But, Mr. B, we CAN do that. We do it all the time." Ok, We'll see, and I turn and write two points on the board... such as (3,1,2) and (2,4,3). They are so sure of their ability that they already have their pencils to paper when they realize they have no idea how to begin. I let them talk, explore, suggest ideas, and I wait, and I wait.... I have never had a student come up with an equation. Some will suggest it must be something like z=ax+by+c or something..... but NOT ONE ever hit upon a correct equation of the line in question.... Then we talk... Not about how to, that will come later, they will discover it on their own as a natural generalization, but about why not.. I admit to them that most of the students who graduate from high school (and in fact, many of the math teachers they have studied under) can not do this simple act of writing the equation of a line in the three space dimension that they live in. And I tell them that in the following weeks they will learn to do some of the simple geometry they know in the dimension they live in. I walk them through a simple vector approach to lines in the coordinate plane. We take y= 3x-1 and rewrite it as (x,y)= (0,-1) + t(1,3). Within minutes every kid in the class can write the equation of a line given two points in this vector form although a few struggle with the seeming reversal of order of the "slope"(in truth, several still mess up regularly when they try to use slope intercept, yet they seem reluctant to adopt the seemingly easier point-slope form). They are quickly taking two points and talking about "point vectors" and "slope vectors" as if they had used them forever. And each year it startles me anew that after a half-hour of an alternate approach, every student will intuitively generalize the method to produce a three-space equation of a line without any help...and then with a little faltering over the "fourth" variable, they can do the same thing in the barely imaginable four-space. Later we will write the equations of planes in space given three points and do some simple analytic geometry in three space. Many of them struggle with the idea of projections of lines and minor details, but I at least feel like I have made a small step to preparing them to function mathematically in the three-space they live in. And if the string-theory guys are right, and we really have a ten-dimensional universe.... no big deal, they can extend vectors to any dimension. But I wonder each year... why are we not introducing this more at an early age (alg I?). I will talk later about some of the advantages I see, and maybe you can tell me what I missing that would make it a bad idea. (Actually in the last few years I've been working a unit in parametric equations and vectors into my Alg II classes with lots of Bowditch curves (Lissajous?) and three D lines and planes and three D geometry ideas.)

I got this comment from one friend...

Here's something marginally related to your topic. In 4 dimensional space it is possible for two planes to have exactly one point in common. A very simple example is the xy-plane, namely all points of the form (x,y,0,0) where x and y vary over the real numbers, and the zw-plane, namely all points of the form (0,0,z,w) where z and w vary over the real numbers. It is easy to see that these two planes have only the origin (0,0,0,0) in common, since x = y = z = w = 0 is the only solution to (x,y,0,0) = (0,0,z,w).

For what it's worth, I first learned about this possibility in a "side-bar diagram" in the following calculus book, which I was trying to work through during the 1974-75 school year (I managed to cover about the first 1/4 of the book), when I was in the 10th grade. Incidentally, the first college math class I took (an ODE course, about a year later) was from one of the authors (Embry) of this text.

Embry/Schell/Thomas, "Calculus and Linear Algebra. An Integrated Approach" (1972)

At this point you might try exploring other possible intersections between two four spaces, intersecting at a single point other than (0,0,0,0) , or a in a line, or a 3-plane. Send me your best ideas, especially for sharing this with students.

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