Scalar or Dot Product

Vectors can be multipled in two different ways to give scalar products or vector products. The scalar product, written A B, also called the dot product or the inner product is equal to a scalar. To see where the scalar product comes from, recall that work in mechanics equals force times displacement. If the force and displacement are not in the same direction, only the component of force along the displacement produces work. We can write w = Fr cos 6, where 6 is the angle between the vectors F and r. 

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For vectors as we have been dealing with, the scalar or dot product is defined as we have seen as follows ... 

The x-component, px, of the momentum now needs to carry a subscript, whereas before it was denoted simply as p. The scalar or dot product of r and p is... 

The unit vectors ei, whose exact definition, meaning and interpretation depend on the particular application at hand, are called basis vectors and form the elements of a basis. They are particularly simple to work with because they are orthogonal. This means that their dot products vanish epej = 0, unless i = j. If i = j, then the scalar or dot product is unity (it is usually convenient, but not necessary, to use bases that are normalized so epei = 1). The shorthand way of representing this information is to write... 

If two vectors C and D are orthogonal, their scalar or dot product will be zero, for... 

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... 

In the absence of Bq, all orientations of the nuclear magnets in space have the same energy. The Bq field, however, interacts with the nuclear magnets to alter their energies as a function of /,. Because energy is a scalar quantity, it is obtained by the scalar or dot product of the magnetic moment and the external magnetic field that is,... 

Prom the geometrical definition of the vector scalar- or dot product between the unit vectors, we can write ... 

The orthogonal curvilinear unit vectors just introduced obay certain laws, which are used in the subsequent paragraphs. The scalar or dot product of two unit vectors yields ... 

Our discussion here refers to vectors in three-dimensional Euclidean space, so vectors are written in one of the equivalent forms a = a, a2,a-i) or ax,aY,az). Two products involving such vectors often appear in our text. The scalar (or dot) product is... 

The scalar or dot product obtained by multiplying two vectors is defined by the operation. 

From vector algebra we recognize that (pv) dA cos a) is the scalar or dot product p(v-n) dA. If we now integrate this quantity over the entire control surface A we have the net outflow of mass across the control surface, or the net mass efflux in kg/s from the entire control volume V ... 

The scalar or dot product processes two vectors, for example v and w, of arbitrary dimension into a scalar. The scalar product is commutative ... 

The binary product in a general vector space is the generalization of the scalar or dot product of two vectors in the normal 3-dimensional vector space ii . 

There are at least two ways to multiply vectors. The iirst way is to find the scalar or dot product between the two vectors. The second way is to find the vector, or cross product, between the two vectors. We shall consider scalar multiplication first. 

Scalar Multiplication. The scalar, or dot product, is defined by the equation... 

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