In this section we review the dot product on vectors. This also includes the angle between vectors and the projection of one vector onto another.
The Dot Product
We begin with the definition of the dot product.
Notice that the dot product takes two vectors and outputs a scalar.
We can also compute the dot product using the magnitude (or length) of the vectors and the angle in between them.
This is illustrated in the picture below.
This provides us with a geometric interpretation of the dot product: it gives us a measure of “how much” in the same direction two vectors are (taking their lengths into account). This also gives us a useful way to compute the angle between two vectors.
From , we then have Solving for , we obtain the angle between the vectors as
Furthermore, note that for nonzero vectors and in , their dot product is if and only if . This means that would have to be or , meaning that and are perpendicular.
This provides us with a very useful algebraic method for determining if two vectors are perpendicular.
By taking the dot product of a vector with itself, we get an important relationship between the dot product and the length of a vector.
This can be shown directly, or using the fact that the angle between and itself is .
Projection of one vector onto another
We can also use the dot product to define the projection of one vector onto another.