2.3 The dot product  (Page 4/16)

In [link] , the direction cosines of $\text{v}=⟨2,3,3⟩$ are $\text{cos}\alpha =\frac{2}{\sqrt{22}},$ $\text{cos}\beta =\frac{3}{\sqrt{22}},$ and $\text{cos}\gamma =\frac{3}{\sqrt{22}}.$ The direction angles of v are $\alpha =1.130\text{rad},$ $\beta =0.877\text{rad},$ and $\gamma =0.877\text{rad}.$

So far, we have focused mainly on vectors related to force, movement, and position in three-dimensional physical space. However, vectors are often used in more abstract ways. For example, suppose a fruit vendor sells apples, bananas, and oranges. On a given day, he sells 30 apples, 12 bananas, and 18 oranges. He might use a quantity vector, $\text{q}=⟨30,12,18⟩,$ to represent the quantity of fruit he sold that day. Similarly, he might want to use a price vector, $\text{p}=⟨0.50,0.25,1⟩,$ to indicate that he sells his apples for 50¢ each, bananas for 25¢ each, and oranges for $1 apiece. In this example, although we could still graph these vectors, we do not interpret them as literal representations of position in the physical world. We are simply using vectors to keep track of particular pieces of information about apples, bananas, and oranges.

This idea might seem a little strange, but if we simply regard vectors as a way to order and store data, we find they can be quite a powerful tool. Going back to the fruit vendor, let’s think about the dot product, $\text{q}·\text{p}.$ We compute it by multiplying the number of apples sold (30) by the price per apple (50¢), the number of bananas sold by the price per banana, and the number of oranges sold by the price per orange. We then add all these values together. So, in this example, the dot product tells us how much money the fruit vendor had in sales on that particular day.

When we use vectors in this more general way, there is no reason to limit the number of components to three. What if the fruit vendor decides to start selling grapefruit? In that case, he would want to use four-dimensional quantity and price vectors to represent the number of apples, bananas, oranges, and grapefruit sold, and their unit prices. As you might expect, to calculate the dot product of four-dimensional vectors, we simply add the products of the components as before, but the sum has four terms instead of three.