While adding vectors to each other and scaling a vector by multiplication by a real number have natural interpretations, we shall now introduce a (first) product of two vectors that is less motivated at first sight, but in fact is very fundamental and useful.
Given two vectors and , we define their scalar product by
We note, as the terminology suggests, that the scalar product of two vectors and is a scalar, that is a number in , while the factors and are vectors in . Note also that forming the scalar product of two vectors involves not only multiplication, but also a summation!
We note the following connection between the scalar product and the norm :
Below we shall define another type of product of vectors where also the product is a vector.
We shall thus consider two different types of products of two vectors, which we will refer to as the scalar product and the vector product, respectively.
At first when limiting our study to vectors in , we may also view the vector product to be a single real number. However, the vector product in is indeed a vector in .
Scalar Product as a Function
We may view the scalar product as a function where .
To each pair of vectors and , we associate the number .
Similarly we may view summation of two vectors as a function .
Here, denotes the set of all ordered pairs of vectors and in .
Example: We have , and $\latex (3,7)\cdot (3,7)=9+49=58$ so that .
Properties of the Scalar Product
The scalar product is linear in each of the arguments and , that is
for all and . This follows directly from the definition. For example, we have
Using the notation , the linearity properties may be expressed as
Scalar Product as Bilinear Form
We also say that the scalar product is a bilinear form on , that is a function from to , since is a real number for each pair of vectors and in and is linear both in the variable (or argument) $\latex a$ and the variable $\latex b$.
Furthermore, the scalar product is symmetric in the sense that
and positive definite in the sense that
We may summarize by saying:
- The scalar product is a bilinear symmetric positive definite form on (with values in .)
We notice that for the basis vectors and , we have
Using these relations, we can compute the scalar product of two arbitrary vectors and in using the linearity as follows:
We may thus define the scalar product by its action on the basis vectors and then extend it to arbitrary vectors using the linearity in each variable.
Scalar Product from Projection
We shall now prove that the scalar product of two vectors and in can be expressed as
- , (1)
where is the angle between the vectors and .
This formula has a geometric interpretation: Assuming that degrees so that is positive, consider the right-angled triangle OAC shown in the figure below.
The length of the side OC is and thus is equal to the product of the lengths of sides OC and OB, if (1) is correct which remains to be shown.
We will refer to OC as the projection of OA onto OB, considered as vectors, and thuswe may say that is equal to the product of the length of the projection of OA onto OB and the length of OB. Because of the symmetry, we may also relate to the projection of OB onto OA, and conclude that is also equal to the product of the length of the projection of OB onto OA and the length of OA:
To prove (1), we write using the polar representation
where is the angle of the direction of and is the angle of direction of . Using a basic trigonometric formula (to be proved in Trigonometric Functions), we see that
where is the angle between and .
Note that since , we may compute the angle between and as or .
We may thus view as the length of the projection of the vector in the direction of , as shown in this figure:
Projection of in the direction of denoted of length
We will come back to the important concept of projection in more detail below.