## Groupings, Shopping Lists, Vectors: part 4

Part 1, part 2, and part 3 of this series looked at a particular model for vector algebra, based on some simple notions such as that of a shopping list.

In this post, I want to play with vector addition and look at the nature of that operation when expressed using the notation developed in the last post.

For a starting point, let’s look at two of your recent gas station receipts:

 9.6 gallons 27.84 dollars

– and –

 8.1 gallons 23.49 dollars

and think about what it would mean to add these up.  The most obvious part of this might be to add up the amounts: \$27.84 + \$23.49 = \$51.33.  What does the amount \$51.33 represent?  It represents the total amount of money you paid to the gas station people for gas, in these two trips to the gas station.

We could also look to see how much gas the \$51.33 paid for.  We got 9.6 gallons on the first trip, and 8.1 gallons on the second trip, for a total of 17.7 gallons.  We can represent the result of our work as follows:

 17.7 gallons 51.33 dollars

and we can consider this our first example of vector addition.  You add the gallons, you add the dollars, and get a new vector, and this new vector still represents what got paid for how much gas – however, this time not through a single purchase, but two purchases combined.

Let’s look at another situation, one where we will look at coins.  Let’s assume that in the ashtray of my car I’ve got the following coins:

 2 quarters 7 pennies 2 nickels

and in my coin purse I’ve got:

 2 dimes 1 Kennedy 50ct piece 3 nickels

and I move all the coins from the ashtray into my coin purse.  What is in my coin purse now?

There is not a lot of overlap between the two: the only kind of coin present in both the ashtray and the coin purse is nickels.  The other coins only occur in one or in the other.  Keeping track of what will be in the coin purse after joining them together is relatively simple.  One way to write the result is:

 2 quarters 7 pennies 5 nickels 2 dimes 1 Kennedy 50ct piece

and you might note that this is a very different answer from \$1.52, which is an answer to a very different question: “what is the value of what is in my coin purse?”

Combining two piles of coins, and keeping track of the total number of each type of coin, that’s a perfect example of vector addition.  (If you’re a math teacher, you might object that my last example isn’t really an example of vector addition, but rather of “adding like terms”.  In response, I’d say that “adding like terms” is a special case of vector addition, and that it is too bad that we usually treat them as completely different and unrelated things.)

Thinking of vectors as piles of stuff where we keep track of what the stuff is and how many of each of the stuff is there – this leads naturally to thinking of vector addition as combining two piles of stuff into a single pile, and continue to keep track of what is there and how many of each are there.