The question in the title came from a fellow teacher who had set in on a class in which I was laying the first foundations of what would be several weeks of focus on vectors. Here is part of an answer:
Teacher: So, you took a year of geometry and two years of algebra, right?
student: That's right, Sir (my kids talk nice)
Teacher: So what can you tell me about any two points in space?
Student: Well, they determine a line, I guess..
Teacher: Good, and if I gave you the coordinates of the two points, could you write the equation of the line?
Student: (enthusiastic now, been here, done this)... Sure!
Teacher: Ok, let one be the point at (1, -2, 7)... and the other be at (3, 1, 5)
Student : (totally confused, now).. Huh... what is that.. you have too many numbers...
Ok, that never happened, but if you want it too, just repeat the teacher part with any bright Pre-calc student and I bet they will fill in the details, maybe even with the "Sir" (or Mam as the case may fit)...
I have wondered for a long time at the overemphasis of the slope-intercept form and the almost total exclusion of functions of more than one variable. So for the next few blogs, I'm going to talk about things you can do easily with vectors that seem difficult or impossible without them (or something like them)...
I think, based on my own experience with students, that it takes very little extra time to take a student from the two dimensional slope interecept form to a vector form of equations that will extend to as many dimensions as the student may ever encounter.. (I assume that number will be finite). Some teachers will suggest that there is not any real difference beween my vector form and what is commonly called parametric form of equations, and I agree, as long as all you want to do is write the equation of a line; but I hope to show in the next few days that mixing vectors, matrices, and the traditional equations of planes can quickly expand the level of three dimenisonal tasks that a student can answer.
Today I want to show that writing the equation of a line with vectors is really as easy as using slope intercept, and may actually make more sense in some ways to the students. Before we jump into three space, we might try to win the student over with a two space example..... for example, suppose we want to write the equation of a line though the points (3,1) and (1,-2) in the X-Y plane. First we find the slope, like always... and it turns out to be 3/2 (what do you say to the kid who writes -3/-2)... Then we can use the slope and one point to write the equation as y-1 = (3/2)(x-2) ...(I walked around an Alg I class today as they were being shown this method, and while the rule and an example were on the board, about 1/3 of the student's whose shoulders I peered over had reversed the x and y coordinate values on the subsequent example ...we drill it into them that x comes before y... hmmm)
I think the kid who really knows what is going on, when asked to graph this line will go to the board, mark the point (3,1) (or perhaps (1,-2)) and then count over two to the right, then up three, and make another point. They might repeat this a couple more times, then sketch a line through the points drawn... To most kids, the line goes "over 2, up three". What if we actually let them write it as [2,3] . [after a couple of examples, we might ask the student how long is the line segment between the two points... wait... see how long it takes them to notice that the two legs are right here in the vector) We could define the line as the set of all points (x,y) so that (x,y)=(3,1) + t [2,3] (there is absolutly no reason mathematically to write the point in parentheses and the slope (vector?) in brackets, but I think there is pedagogically). Now imagine that a classroom full of kids had been trained to write the two-dimensional equations as shown, and then we say... hey guys, suppose we want to write the equation of two points in space, with points (x,y,z)... between the two points at (1, -2, 7)... and at (3, 1, 5). I can tell you that in my experience working with Alg II and Pre-calc kids, they automatically extend the method naturally..... almost every one of them... and when I go really crazy, and ask them to write a line between two points in four space, over half the kids in class are wagging hands to get a shot at the board. And when I ask them the distance between the two points, there is almost never a question about whether the Pythagorean method applies... vectors is vectors, baby!
Down the line, I think there are some great modifications. If we develop the practice of writing the slope vector (I call it that becuase they learned "slope" first, but it could easily be called the translation vector) as a unit vector. The variable t suddenly takes on the additional value of indicating the distance away from the original point in the vector direction; substitute in t=3, and you get the point 3 units away..etc. Now we ask them to find the point three-fifths of the way from point A toward point B and they write A + 3/5 [B-A] almost without instruction; and yes, they will know what to do if you ask for it the other way around.
There are some heavy ideas you might want to explore here that you could almost never touch using the traditional equations. Imagine giving your students the coordinates of a triangle in three space and have the students find the point where the medians intersect. Later with dot and vector products, and a little work with matrices to find solutions to systems of equations, we will find the equation of a line along the intersection of two planes, the foot of the perpendicular to a tetrahedron etc.
Someone who is really good at linear algebra could probably point out fifty other little tasks that are easy to do in three-space with vectors and matrices.. (and I would love to hear from you)... But for now, try introducing some vector equations of lines to your kids.. I bet they can pick it up in one class period, and it is a natural companion to parametric equations (which is the area of the curriculum I use to justify the very little time I go off task on three space vectors and matrices)
Stay tuned... Next I'll write use the dot product to find the angle between two lines or segments in space.
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