The previous installments in this series can be found here and here.

What I am after is a natural way to develop the usefulness of the idea of a vector, and in the previous post I suggested that a shopping list gives us a good start.  Here, I want to show some natural notations for that idea, even though that notation will look very different from the standard notation used in typical math class and typical textbooks.

Let’s start with this:

You can recognize it as a shopping list for a grocery store.  We want 5 apples, 2 six-packs of Coke, etc.  For each item, we say what it is we want, and how many of those we want.  Though it looks like a bunch of different things that are only loosely connected, it is essential for our purposes here that we can also look at it as a single whole.  That single whole is the shopping list, and it corresponds to a single trip to the grocery store, a single shopping cart, and a single grocery receipt.  The single whole is what we are calling a vector, here represented as a box with two columns, one column for the things and one column to indicate how many of each of these things.

Sometimes, we will find it useful to turn the whole thing on its side:

The standard notation for vectors in textbooks would show this as (5,2,3,2), which leaves out the very things that allows us to see what the numbers are supposed to mean.  The good thing about the standard notation, of course, is that it is really really compact.  But sometimes being too concise can remove all the flavor.  (There is the famous joke from Woody Allen where he describes War and Peace after speed-reading it: “It’s about some Russians.”)

Of course, there are other situations than shopping lists that could be modeled in the same way.  Here, as an example, is a stock portfolio:

The box above represents a single portfolio, consisting of three different stocks.  It isn’t enough to know which stocks you own, it also matters how many shares of each.

And the “X” on the pirate’s treasure map may indicate the location of the treasure like so:

and the readout on the gasoline pump (and on your receipt) might say:

The position of a ball dropped from a height on a graph might be recorded as:

So what’s the big deal?  We haven’t done anything yet, we have just recorded some data in a consistent format.  All I can suggest at this point is that this consistent format is convenient and somewhat self-contained.  In the next post I will introduce simple operations on vectors, such as vector addition.

Coming Attractions: soon, we’ll even make sense out of vector inner products and show how those arise in a natural way.