Scalar product of two vectors

As the scalar product of two vectors is related to the cosine of the angle included by these vectors by Eq. (4), a frequently used similarity measure is the cosine coefficient (Eq. (5)). 

The product of two matrices is therefore similar to the scalar product of two vectors. C is the product of AB, according to... 

The ordinary three-dimensional space of position vectors is also an inner product space with the familiar rule for taking the scalar product of two vectors. 

Such a quantity has here been called an invariant or a scalar. The scalar product of two vectors is a contracted tensor, AVBV = (hu/hu) AUBV1 and is, therefore an invariant. 

Eventually, quantities D can be considered as the Ath element of vector 5, so we can write the element of two-electron density matrix 0 as a scalar product of two vectors ... 

There is an equivalent but more generally useful way of writing the scalar product of two vectors. Suppose that we have two vectors A and B, both lying in the xy plane. Let A make an angle (f> to the x axis and B a greater... 

This may be considered a scalar product of two vector operators L1 and S1 (the first-rank tensors)... 

Note that when we come to consider terms in this Hamiltonian such as S, (r ji a />, ) which represents the scalar product of two vectors that are defined in different coordinate systems, we must necessarily transform one operator to the coordinate system of the other before we can evaluate these terms. 

This differs from equation (5.110) in both phase and normalisation factors. We have seen that, for k = 1, the spherical tensor corresponds to a cartesian vector the spherical scalar product in this case is the same as the cartesian scalar product of two vectors ... 

As the name implies, the scalar product is a way of multiplying vectors which results in a scalar quantity. It is also known as the dot product, because the multiplication operation is represented by a dot. The scalar product of two vectors a and b is defined by... 

Because of the orthogonality of the unit vectors in a scalar or dot product, the scalar product of two vectors A and B can be expressed as... 

Geometrically, Fig. A1-1 shows that the scalar product of two vectors may be regarded as the product of the length of one vector and the projection of the other upon the first. If one of the vectors, say a, is a unit vector (a vector of unit length), then a b gives immediately the length of the projection of b on a. The scalar product of sums or differences of vectors is formed simply by term-by-term multiplication ... 

The scalar product of two vectors, u and v, may be written as n%. Two vectors are orthogonal if their scalar product is zero. A particularly important quantity involving the scalar product is the Euclidean norm of a vector defined by u = (u u). The Euclidean norm of a vector is non-negative and is zero only if the vector is zero. 

At this point it is convenient to introduce two other vector operations, namely, the scalar product and the vector product. The scalar product of two vectors /and g is denoted a.sf-g and is given by... 

The scalar product of two vectors a and A, expressed In terms of base vectors, is obtained by taking the sum of the scalar products of each base vector pair, together with the appropriate product of components. 

We next define the scalar product of two vectors, which is also called the dot product because of the use of a dot to represent the operation. If A and B are two vectors, and a is the angle between them, their scalar product is denoted by A B and given by... 

When the g-tensor components are distinguished, one can use the fact that the scalar product of two vectors can be written in terms of the spherical tensor components, hence... 

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