When dealing with 2- and 3-dimensional vectors in Euclidean Space, as we have been doing all along, different methods of vector multiplication can be very helpful. The notions of vector multiplication we will define allow us to extract useful geometric information about our vectors.
The first type of vector multiplication we will discuss is called the dot product. The dot product involves multiplying two vectors together to get a scalar, not another vector (for this reason, the dot product is often referred to as a scalar product). We will use the dot product to obtain information about the length (or magnitude) of vectors, as well as to compute the degree to which two vectors "overlap." We will define the dot product in both the 2- and 3-dimensional cases.
The second kind of vector multiplication we will find useful is called the cross product. Contrary to the dot product, the cross product multiplies two vectors together to obtain a third vector rather than a scalar. However, we will only be able to define the cross product in the case of 3-dimensional vectors. There is no cross product in the 2-dimensional case.