Intuition for using vectors in sale related problems

Question

I am reading Linear Algebra from David Lay's book. He gives one example to showcase use of linear combination of vectors :

I understand the solution, but I am completely clueless about how to intuitively think of using vectors in a problem like this ? I could find no indication that vector as a concept would be helpful in this case ? How am supposed to think of quantities like cost/voltage etc. as vectors ?

Randomly given these kind of problems, what hint should direct me to use vectors ?

Is my prior understanding of vectors in terms of Physics causing this problem ? (I still have that notion of quantity having a direction for it to be vector)

Answer 1

Well I think first of all you have to think about vectors as containers that hold data rather than thinking the physics kind of way. Vectors are used very often in computer architecture to carry values and use these values when the computer needs them.

Lets say you want to calculate the price of some groceries

you have apples $3 , oranges $5, bananas $2 (each price is for one item). These are set prices. and you have a customer that purchased 5 apples 3 oranges and no bananas. you need to calculate the total price.

so you have: 3$ * 5 + $5 * 3 + $2 * 0.

OR YOU COULD ARRANGE THEM IN VECTORS AND USE A SCALAR PRODUCT...

scalar product of 2 vectors lets say price = [3,5,2] and quantity = [5,3,0]

price.quantity = $3*5 + $5*3 + $2 * 0

http://en.wikipedia.org/wiki/Dot_product

Answer 2

Is my prior understanding of vectors in terms of Physics causing this problem?

Probably, but rightfully so.

In this case the components of the vector don't reflect a physical space of some kind.

However, the components of the vector aren't that conceptually abstract as to fall under how a programmer might use them in object-oriented programming.

The key, I think, is that all of the components of the vector have a meaningful representation under operations you'd see in a vector space.

In other words, $100$ times this vector actually means something. The sum of $b$ and $c$ actually means something.

We could also add some other component to the vector -- even if it's not a cost -- as long as it behaves like the other component. Let's say each $1 of product fits in a box, and that's how they store it in their warehouse.

Then we could have

$$b' = \left[\begin{matrix}.45 \\ .25 \\ .15 \\ 1\end{matrix}\right]$$

The top three are costs, in dollars, and the bottom one is storage, in boxes.

However, this doesn't work for everything. What about square footage to store the boxes? This might break the meaning, because $\$1,000$ worth of product doesn't take up $1000$ times the square footage to store.

Less contrived ways that vectors are used with different kinds of components are to describe a generalized state of an object. Navigation uses this often. You can specify positions, velocities, accelerations, time biases, and lots more in a single state vector and pass the vector through matrix operations to arrive at a solution.