The Dot Product

The dot product is one of two ways that we can multiply two vectors. The result of the dot dot product is a scalar quantity. Thus, we start with two vectors quantities, say and and form their dot product, written as . The answer will be a number that, in some sense, tells us how similar the two vectors are. The more positive the answer is, the closer the two vectors are in terms of their directions.

One way to calculate the dot product of two vectors is shown below.

The dot product is, more or less, the component of the first vector along the direction of the second vector. Simple trigonometry gives us that this should be the same as where is the angle between the two vectors. In truth, the dot product also includes the length of the other vector, so that

Now, it is easy to see that if the two vectors have identical directions, then the angle between the vectors is 0, so the dot product will be a maximum. At what angles would the dot product vanish? Watch the animation below to see. In the illustration, both vectors have unit length for simplicity.

Let's use these ideas to come up with a few simple dot products.

1. Let's start with an easy one. Since the length of the vector is 1, we know that because the angle between the vectors is 0, making the cosine of the angle 1. In fact, the dot product of any unit vector with itself should be 1. Check this for and .
2. What happens if we dot the unit basis vectors any other way? Try it with and . Since the angle between the vectors is 90 degrees, the cosine of the angle is 0 so the dot product must also be zero.
3. Now let's get away from the unit vectors.

   

That's not so bad. But what about more general vectors that don't point directly along the basis vectors? Fortunately for us, there are properties of the dot product that help us out.