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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/Courses/Lorain_County_Community_College/Book%3A_Precalculus_(Stitz-Zeager)_-_Jen_Test_Copy/11%3A_Applications_of_Trigonometry/11.09%3A_The_Dot_Product_and_ProjectionPreviously, we learned how add and subtract vectors and how to multiply vectors by scalars. In this section, we define a product of vectors.
- https://math.libretexts.org/Bookshelves/Calculus/Map%3A_University_Calculus_(Hass_et_al)/11%3A_Vectors_and_the_Geometry_of_Space/11.3%3A_The_Dot_Product\[ \begin{align*} ‖\vecs{ v}−\vecs{ u}‖^2 &=(\vecs{ v}−\vecs{ u})⋅(\vecs{ v}−\vecs{ u}) \\[4pt] &=(\vecs{ v}−\vecs{ u})⋅\vecs{ v}−(\vecs{ v}−\vecs{ u})⋅\vecs{ u} \\[4pt] &=\vecs{ v}⋅\vecs{ v}−\vecs{ u...\[ \begin{align*} ‖\vecs{ v}−\vecs{ u}‖^2 &=(\vecs{ v}−\vecs{ u})⋅(\vecs{ v}−\vecs{ u}) \\[4pt] &=(\vecs{ v}−\vecs{ u})⋅\vecs{ v}−(\vecs{ v}−\vecs{ u})⋅\vecs{ u} \\[4pt] &=\vecs{ v}⋅\vecs{ v}−\vecs{ u}⋅\vecs{ v}−\vecs{ v}⋅\vecs{ u}+\vecs{ u}⋅\vecs{ u} \\[4pt] &=\vecs{ v}⋅\vecs{ v}−\vecs{ u}⋅\vecs{ v}−\vecs{ u}⋅\vecs{ v}+\vecs{ u}⋅\vecs{ u} \\[4pt] &=‖\vecs{ v}‖^2−2\vecs{ u}⋅\vecs{ v}+‖\vecs{ u}‖^2.\end{align*}\]
- 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/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/06%3A_Orthogonality/6.01%3A_Dot_Products_and_OrthogonalityThis page covers the concepts of dot product, vector length, distance, and orthogonality within vector spaces. It defines the dot product mathematically in \(\mathbb{R}^n\) and explains properties lik...This page covers the concepts of dot product, vector length, distance, and orthogonality within vector spaces. It defines the dot product mathematically in \(\mathbb{R}^n\) and explains properties like commutativity and distributivity. Length is derived from the dot product, and the distance between points is defined as the length of the connecting vector. Unit vectors are introduced, and orthogonality is defined as having a dot product of zero.
- 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/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/Calculus/Book%3A_Active_Calculus_(Boelkins_et_al.)/09%3A_Multivariable_and_Vector_Functions/9.03%3A_Dot_ProductIn this section, we will introduce a means of multiplying vectors.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Understanding_Linear_Algebra_(Austin)/06%3A_Orthogonality_and_Least_Squares/6.01%3A_The_dot_productIdentify the vectors \(\mathbf p\) and \(\mathbf n\) for the line illustrated in Figure 6.1.16 and use them to write the equation of the line in terms of \(x\) and \(y\text{.}\) Verify that this expre...Identify the vectors \(\mathbf p\) and \(\mathbf n\) for the line illustrated in Figure 6.1.16 and use them to write the equation of the line in terms of \(x\) and \(y\text{.}\) Verify that this expression is algebraically equivalent to the equation \(y=mx+b\) that you earlier found for this line.
- https://math.libretexts.org/Courses/Lorain_County_Community_College/Book%3A_Precalculus_Jeffy_Edits_3.75/11%3A_Applications_of_Trigonometry/11.09%3A_The_Dot_Product_and_ProjectionPreviously, we learned how add and subtract vectors and how to multiply vectors by scalars. In this section, we define a product of vectors.
