INSTRUCTOR: AAA Party Supply Store sells invitations, party favors, decorations, and food service items, such as paper plates and napkins. When AAA buys inventory, it pays $0.25 per package for invitations and party favors. Decorations cost AAA $0.50 each. And food service items cost $0.20 per package.

Triple AAA sells invitations for $2.50 per package and party favors for $1.50 per package. Decorations sell for $4.50 each. And food service items for $1.25 per package.

Now, this is all previous information we're going to need in this question. So now for the question at hand. On June 1, AAA Party Supply Store decided to increase the price they charge for party favors to $2 per package. They also changed suppliers for their invitations and are now able to purchase invitations for only $0.10 per package.

All their other costs and prices remain the same. If AAA sells 1,408 invitations, 147 party favors, 2,112 decorations, and 1,894 food service items in the month of June, use vectors and dot products to calculate their total sales and profit for June.

Now first, I'm going to write several vectors. One is the cost vector. So c, I'm going to write this as the vector. And let's see. I'm going to go in order here. This will be invitations will be my first component.

My second component will be party favors. I'll abbreviate that as PF. Decorations for my third component, and food service or FS for my final component.

So the cost of invitations was $0.25 per package, now is $0.10 per package. So 0.1 is the first component of the cost vector. Party favors did not change. That's still 0.25 in the second component. Decorations are-- that costs $0.50 per package, so 0.5. And food service items for my final component is 0.2.

Next I want to write down a vector for prices-- the prices they are selling them at. And it looks like $2.50-- yup, for the invitations. $2.50, so 2.5 my first component. Again, lining up with invitations. Now for party favors, they've raised their prices to $2. Decorations are $4.50. And food service items are 1.25 for $1.25. And that is my price vector.

Finally, we have some information that's quantities. These are the amount they're selling. Amount they've sold of invitations is 1,408, party favors is 147, decorations are 2,112, and food service items 1,894. That is my final vector.

Now thinking about financial math here, to find the total sales, that's like the revenue. It's the revenue. So this is going to be the price they sell items at. The price they sell items at, vector p.

And typically, we would multiply this by the quantity that they sell. Well in this case, since we're extending this into a third dimension here, we've got the dot product. So the vector p dot vector q for our quantities.

So that would mean we have 2.5 multiplied by 1,408, plus 2 multiplied by 147 plus 4.5 multiplied by 2,112, plus 1.25 multiplied by 1,894. So that my total sales is going to be $15,685.50.

That's our total sales. Now next we want to find profit. Now there are two ways we can find profit. We can take their costs and subtract that from the sales. That would be one way. Or you can use a little bit of vector arithmetic. I'll actually show you what that'll be really quickly.

So my end result is going to be p dot q minus c dot q, c dot q being the cost-- what it costs them to purchase these things. Which that would then be equal to quantity p minus c dotted with q. And that would actually give us our final answer. So that's one way to do this.

I'm going to go at this a little bit more directly. But I just want to point out that you could actually do that. Vector arithmetic is nice sometimes.

So let's go ahead and find their costs. So I'm going to abandon that method right now. Their costs would be vector c dotted with q, the cost per item multiplied by the quantities. Now using the two vectors we have up here, this will be 0.1 multiplied by 1,408, plus 0.25 multiplied by 147, plus 0.5 multiplied by 2,112, plus 0.2 multiplied by 1,894.

That sum would be a total cost of $1,612.35. So the profit, which is the value we actually want, is going to be $15,685.50 minus $1,612.35, which is $14,073.15.