The Scalar, Dot, or Inner Product of Two Vectors

The dot or inner product of two vectors is a scalar quantity and is, in matrix notation. 

The inner or dot product of two vectors is defined only between a row vector x and a column vector y, and only when the number of columns in is the same as the number of rows in y. The dot product is a scalar given by... 

The operation of multiplication between the two vectors U/, and dr is the dot product (or scalar product or inner product), which is zero when the two vectors are orthogonal and maximum when they are parallel. 

The double contracting or double inner product of general tensors results in a tensor with the added order of the multiplied tensors lowered by four. The employed symbol of two dots alludes to the two scalar products of the particular base vectors. In the case of two tensors of second order, the outcome is of zeroth order, leading to the denomination as a scalar product of dyads. The double contracting product is commutative, given here for the case of dyads ... 

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. 

This notation is analogous to the dot product used in the analysis of vectors in Euclidean space. The dot product is an operation that maps two vectors into a scalar. Here, in the context of functions, the inner product defined in Eqs. 8.7 or 8.8a will map two functions into a scalar, which will be used in the process of minimization of the residual R. This minimization of the residual intuitively implies a small error in the approximate solution, y (x). 

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