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- https://math.libretexts.org/Courses/Coastline_College/Math_C285%3A_Linear_Algebra_and_Diffrential_Equations_(Tran)/04%3A_R/4.07%3A_The_Dot_ProductThere are two ways of multiplying vectors which are of great importance in applications. The first of these is called the dot product. When we take the dot product of vectors, the result is a scalar. ...There are two ways of multiplying vectors which are of great importance in applications. The first of these is called the dot product. When we take the dot product of vectors, the result is a scalar. For this reason, the dot product is also called the scalar product and sometimes the inner product.
- https://math.libretexts.org/Bookshelves/Differential_Equations/Differential_Equations_for_Engineers_(Lebl)/4%3A_Fourier_series_and_PDEs/4.02%3A_The_trigonometric_seriesThe document discusses the concept of periodic functions, focusing on Fourier series as a method to solve differential equations with periodic inputs. It provides a step-by-step explanation of how to ...The document discusses the concept of periodic functions, focusing on Fourier series as a method to solve differential equations with periodic inputs. It provides a step-by-step explanation of how to decompose a periodic function into a sum of sine and cosine terms, leveraging the orthogonality of trigonometric functions to determine expansion coefficients. Examples illustrate how functions like the sawtooth and square wave can be expressed as Fourier series.
- https://math.libretexts.org/Courses/Mission_College/Math_4A%3A_Multivariable_Calculus_v2_(Reed)/12%3A_Vectors_in_Space/12.03%3A_The_Dot_ProductIn this section, we develop an operation called the dot product, which allows us to calculate work in the case when the force vector and the motion vector have different directions. The dot product es...In this section, we develop an operation called the dot product, which allows us to calculate work in the case when the force vector and the motion vector have different directions. The dot product essentially tells us how much of the force vector is applied in the direction of the motion vector. The dot product can also help us measure the angle formed by a pair of vectors and the position of a vector relative to the coordinate axes.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Matrix_Algebra_with_Computational_Applications_(Colbry)/35%3A_18_Pre-Class_Assignment_-_Inner_Product/35.1%3A_Inner_ProductsDefinition: An inner product on a vector space \(V\) (Remember that \(R^n\) is just one class of vector spaces) is a function that associates a number, denoted as \(\langle u,v \rangle\), with each pa...Definition: An inner product on a vector space \(V\) (Remember that \(R^n\) is just one class of vector spaces) is a function that associates a number, denoted as \(\langle u,v \rangle\), with each pair of vectors \(u\) and \(v\) of \(V\). Two vectors \(u\) and \(v\) in \(V\) are orthogonal if their inner product is zero: The distance between two vectors (points) \(u\) and \(v\) in \(V\) is denoted by \(d(u,v)\) and is defined by:
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Linear_Algebra_with_Applications_(Nicholson)/10%3A_Inner_Product_Spaces/10.01%3A_Inner_Products_and_NormsThe dot product was introduced in \(\mathbb{R}^n\) to provide a natural generalization of the geometrical notions of length and orthogonality that were so important in Chapter 4 . The plan in this cha...The dot product was introduced in \(\mathbb{R}^n\) to provide a natural generalization of the geometrical notions of length and orthogonality that were so important in Chapter 4 . The plan in this chapter is to define an inner product on an arbitrary real vector space \(V\) (of which the dot product is an example in \(\mathbb{R}^n\) ) and use it to introduce these concepts in \(V\).
- https://math.libretexts.org/Under_Construction/Purgatory/MAT-004A_-_Multivariable_Calculus_(Reed)/01%3A_Vectors_in_Space/1.04%3A_The_Dot_ProductIn this section, we develop an operation called the dot product, which allows us to calculate work in the case when the force vector and the motion vector have different directions. The dot product es...In this section, we develop an operation called the dot product, which allows us to calculate work in the case when the force vector and the motion vector have different directions. The dot product essentially tells us how much of the force vector is applied in the direction of the motion vector. The dot product can also help us measure the angle formed by a pair of vectors and the position of a vector relative to the coordinate axes.
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Applied_Geometric_Algebra_(Tisza)/01%3A_Algebraic_Preliminaries/1.04%3A_How_to_multiply_vectors_Heuristic_considerationsIn evaluating the various methods of multiplying vectors with vectors, we start with a critical analysis of the procedure of elementary vector calculus based on the joint use of the inner or scalar pr...In evaluating the various methods of multiplying vectors with vectors, we start with a critical analysis of the procedure of elementary vector calculus based on the joint use of the inner or scalar product and the vector product. Although it is possible to adapt quaternions to deal with the Lorentz group, it is more practical to use instead the algebra of complex two-by-two matrices, the so-called Pauli algebra, and the complex vectors (spinors) on which these matrices operate.
- https://math.libretexts.org/Courses/Lake_Tahoe_Community_College/Interactive_Calculus_Q3/04%3A_Vectors_in_Space/4.04%3A_The_Dot_ProductIn this section, we develop an operation called the dot product, which allows us to calculate work in the case when the force vector and the motion vector have different directions. The dot product es...In this section, we develop an operation called the dot product, which allows us to calculate work in the case when the force vector and the motion vector have different directions. The dot product essentially tells us how much of the force vector is applied in the direction of the motion vector. The dot product can also help us measure the angle formed by a pair of vectors and the position of a vector relative to the coordinate axes.
- https://math.libretexts.org/Courses/City_College_of_San_Francisco/CCSF_Calculus/12%3A_Vectors_in_Space/12.04%3A_The_Dot_ProductIn this section, we develop an operation called the dot product, which allows us to calculate work in the case when the force vector and the motion vector have different directions. The dot product es...In this section, we develop an operation called the dot product, which allows us to calculate work in the case when the force vector and the motion vector have different directions. The dot product essentially tells us how much of the force vector is applied in the direction of the motion vector. The dot product can also help us measure the angle formed by a pair of vectors and the position of a vector relative to the coordinate axes.
- https://math.libretexts.org/Bookshelves/Differential_Equations/A_Second_Course_in_Ordinary_Differential_Equations%3A_Dynamical_Systems_and_Boundary_Value_Problems_(Herman)/06%3A_Sturm_Liouville/6.02%3A_Properties_of_Sturm-Liouville_Eigenvalue_ProblemsThere are several properties that can be proven for the (regular) SturmLiouville eigenvalue problem. However, we will not prove them all here. We will merely list some of the important facts and focus...There are several properties that can be proven for the (regular) SturmLiouville eigenvalue problem. However, we will not prove them all here. We will merely list some of the important facts and focus on a few of the properties.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/07%3A_Vector_Spaces/7.12%3A_Inner_Product_Spaces/7.12E%3A_Exercises_for_Section_7.12This page contains exercises on inner product spaces, focusing on identifying properties and verifying definitions. Key activities include analyzing axioms for inner products, demonstrating subspaces,...This page contains exercises on inner product spaces, focusing on identifying properties and verifying definitions. Key activities include analyzing axioms for inner products, demonstrating subspaces, computing distances, and checking properties for functions in \(D_n\). It emphasizes symmetry, linearity, and positive-definiteness while exploring inner products in complex numbers and matrices.
