In this section we’re going to be taking a look at two special products of vectors, the dot product and the cross product.  However, before we look at either on of them we need to get a quick definition out of the way.

Suppose that u and v are two vectors in 2-space or 3-space that are placed so that their initial points are the same.  Then the angle between u and v is angle  that is formed by u and v such that .  Below are some examples of angles between vectors.

Notice that there are always two angles that are formed by the two vectors and the one that we will always chose is the one that satisfies .  We’ll be using this angle with both products.

So, let’s get started by taking a look at the dot product.  Of the two products we’ll be looking at in this section this is the one we’re going to run across most often in later sections.  We’ll start with the definition.

Note that the dot product is sometimes called the scalar product or the Euclidean inner product.  Let’s see a quick example or two of the dot product.

Now, there should be a question in everyone’s mind at this point.  Just how did we arrive at those angles above?  They are the correct angles, but just how did we get them?  That is the problem with this definition of the dot product.  If you don’t have the angles between two vectors you can’t easily compute the dot product and sometimes finding the correct angles is not the easiest thing to do.

Fortunately, there is another formula that we can use to compute the formula that relies only on the components of the vectors and not the angle between them.

Proof : We’ll just prove the 3-space version of this theorem.  The 2-space version has a similar proof.  Let’s start out with the following figure.

So, these three vectors form a triangle and the lengths of each side is , , and .  Now, from the Law of Cosines we know that,

Now, plug in the definition of the dot product and solve for .

Next, we know that  and so we can compute .  Note as well that because of the square on the norm we won’t have a square root.  We’ll also do all of the multiplications.

The first three terms of this are nothing more than the formula for  and the next three terms are the formula for .  So, let’s plug this into (1).

And we’re done with the proof.

Before we work an example using this new (easier to use) formula let’s notice that if we rewrite the definition of the dot product as follows,

we now have a very easy way to determine the angle between any two vectors.  In fact this is how we got the angles between the vectors in the first example!