If we draw both vectors separated by the angle and then try to find the image of the scalar product, we will realize that this consists of the multiplication of two parts: the projection of one vector to the direction of the second one and the same but for the second vector. As a conclusion of this section, we can say that the dot product is a multiplication of a vectors' lengths, projected in the direction of one of the other.
With this property, the inner product is called a Hermitian inner product and a complex vector space with a Hermitian inner product is called a Hermitian inner product space. As we know from the previous section, it's the projection of a to the direction of d multiplied by d.
Related thereto, note that some authors define an inner product to be a function satisfying only the first three of the above conditions with the added weaker condition of being weakly non-degenerate i. We can define a triple product or mixed product as a combination of a dot product and a vector product.
In this case, the dot product is used for defining lengths the length of a vector is the square root of the dot product of the vector by itself and angles the cosine of the angle of two vectors is the quotient of their dot product by the product of their lengths.
The last step is obviously possible because the multiplication of lengths is commutative. In such literature, functions satisfying all four such conditions are typically referred to as positive-definite inner products Ratcliffethough inner products which fail to be positive-definite are sometimes called indefinite to avoid confusion.