I think when most of us introduce vectors, we mostly focus on the differences between scalars and vectors. I try to convince students that almost everything they know about numbers applies to vectors. I tell them this because there are so many good things they can bring over from numbers, and only a relatively few number of ideas that can go awry. Here are some good things you want your students to bring over from numbers to vectors
The orientation of symbols that represent numbers (or vectors) are important because they tell you about their meaning
For example, 6 and 9 are not the same symbol. Of course, students exploit this property of number symbols all the time to write funny words in calculators.
Symbols that represent numbers (or vectors) can put into new arrangements in order to make it easier to carryout certain procedures with them
For example, 71 + 82 is the same as 71
When we do this, the meaning of 71, 82, and + don’t change. It simply allows us see how place values compare so that one procedure for adding becomes easier to carry out.
Symbols that refer to “whole” can often be described in terms of “various parts”
987 = 9 x 10² + 8 x 10¹ + 7 x 10°
987 = 900 + 87
987 = 460*2 + 67
This is something students struggle with even with numbers–decomposition and re-composition. So it’s not a surprise it’s hard for students to understand vector components, and odd things like vector components with tilted axes.
Getting back to Vectors
While it’s true that all these properties are also important with vectors, instruction with vectors often makes it seem like these things are special, unique, and weird about vectors. They emphasize how you must carefully move vectors without rotating them. They make a big deal about how you move vectors to carryout procedures of adding them. They make a big deal about how you can write vectors in terms of components.
I’m not saying that students don’t have to learn some new ideas, procedures, and to learn to distinguish scalars from vectors. I’m just saying let’s not pretend that vectors are all that different than scalars. The difference is subtle, somewhat like the difference between a square and a rectangle. Both squares and rectangles are quadrilaterals. They have a lot more in common than different.
Magnitude and Direction?
Another interesting thing about how we teach vectors comes from something I recently re-read in A. Arons’, “Teaching Introductory Physics”. Arons reminds us that we teach most students that vectors are “magnitude and direction”. Arons points out that vectors commute upon addition, which is something also true for numbers. But Arons reminds us that this is not true for all things with “magnitude and direction”. Everyone familiar with rotations knows that finite angular displacement have both magnitude and direction, but that they don’t generally commute upon successive displacement. Arons points out that “magnitude and direction” is not only a wrong definition for vector. Nor is it just an incomplete definition. He seems to suggest that it is misleading and possibly not generative for later learning. He proposes that it should be emphasized that anything that commutes is a vector. This, of course, make scalars a kind of vector, which is actually how we kind of think about scalars as zeroth order (or rank) tensors. Scalars are a special kind of vector (or tensor) that doesn’t transform under rotation. This makes the analogy about squares, rectangles, and quadrilaterals a little closer to the truth.
The comment about ‘all things are vectors’ is particularly appealing to me. Its amazing how many physics majors, grads and undergrads think about vectors as pointy arrows instead of elements of a vector space. I find the ‘direction and magnitude’ definition silly especially considering wave functions are vectors in a Hilbert space and I’d like to see someone “point” in the direction of psi. Furthermore, vectors only have magnitude if we have a metric tensor an inner product on the space, which is not guaranteed. And as you say, thinking of everything as pointy arrows is also silly when you generalize, not everything transforms as a pointy arrow (thinking geometrically).