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- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Variational_Principles_in_Classical_Mechanics_(Cline)/19%3A_Mathematical_Methods_for_Classical_Mechanics/19.03%3A_Appendix_-_Vector_algebraScalar, vector, tensor products of linear operators.
- https://phys.libretexts.org/Courses/Prince_Georges_Community_College/General_Physics_I%3A_Classical_Mechanics/10%3A_The_Dot_ProductWe begin to look at vector multiplication starting in this chapter with the dot product, sometimes called the scalar product because it multiplied two vectors together to yield a scalar.
- https://phys.libretexts.org/Courses/Georgia_State_University/GSU-TM-Physics_II_(2212)/01%3A_Introduction_to_Physics_Measurements_and_Mathematics_Tools/1.08%3A_Vectors/1.8.06%3A_Products_of_Vectors\[\begin{split} \vec{A} \times \vec{B} & = (A_{x}\; \hat{i} + A_{y}\; \hat{j} + A_{z}\; \hat{k}) \times (B_{x}\; \hat{i} + B_{y}\; \hat{j} + B_{z}\; \hat{k}) \\ & = A_{x}\; \hat{i} \times (B_{x}\; \ha...\[\begin{split} \vec{A} \times \vec{B} & = (A_{x}\; \hat{i} + A_{y}\; \hat{j} + A_{z}\; \hat{k}) \times (B_{x}\; \hat{i} + B_{y}\; \hat{j} + B_{z}\; \hat{k}) \\ & = A_{x}\; \hat{i} \times (B_{x}\; \hat{i} + B_{y}\; \hat{j} + B_{z}\; \hat{k}) + A_{y}\; \hat{j} \times (B_{x}\; \hat{i} + B_{y}\; \hat{j} + B_{z}\; \hat{k}) + A_{z}\; \hat{k} \times (B_{x}\; \hat{i} + B_{y}\; \hat{j} + B_{z}\; \hat{k}) \\ & = A_{x}B_{x}\; \hat{i} \times \hat{i} + A_{x}B_{y}\; \hat{i} \times \hat{j} + A_{z}B_{z}\; \ha…
- https://phys.libretexts.org/Courses/Joliet_Junior_College/Physics_201_-_Fall_2019v2/Book%3A_Custom_Physics_textbook_for_JJC/03%3A_Vectors/3.08%3A_Products_of_Vectors_(Part_1)One kind of vector multiplication is the scalar product, also known as the dot product, which results in a number (scalar). The scalar product has the distributive property and the commutative propert...One kind of vector multiplication is the scalar product, also known as the dot product, which results in a number (scalar). The scalar product has the distributive property and the commutative property, and is obtained by multiplying the magnitudes of the two vectors with the cosine of the angle between them. This type of vector multiplication is used to find angles between vectors and in the definitions of derived scalar physical quantities such as work or energy.
- https://phys.libretexts.org/Courses/Prince_Georges_Community_College/General_Physics_I%3A_Classical_Mechanics/10%3A_The_Dot_Product/10.01%3A_DefinitionThe dot product of two vectors \(\mathbf{A}\) and \(\mathbf{B}\) (written \(\mathbf{A} \cdot \mathbf{B}\), and pronounced "A dot \(\mathbf{B}\) ") is defined to be the product of their magnitudes, tim...The dot product of two vectors \(\mathbf{A}\) and \(\mathbf{B}\) (written \(\mathbf{A} \cdot \mathbf{B}\), and pronounced "A dot \(\mathbf{B}\) ") is defined to be the product of their magnitudes, times the cosine of the angle between them: It turns out that this combination occurs frequently in physics; the dot product is related to the projection of one vector onto the the other.
- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Classical_Mechanics_(Dourmashkin)/13%3A_Energy_Kinetic_Energy_and_Work/13.08%3A_Work_and_the_Scalar_ProductThe scalar product \(\overrightarrow{\mathbf{A}} \cdot \overrightarrow{\mathbf{B}}\) of the vectors \(\overrightarrow{\mathbf{A}}\) and \(\overrightarrow{\mathbf{B}}\) is defined to be product of the ...The scalar product \(\overrightarrow{\mathbf{A}} \cdot \overrightarrow{\mathbf{B}}\) of the vectors \(\overrightarrow{\mathbf{A}}\) and \(\overrightarrow{\mathbf{B}}\) is defined to be product of the magnitude of the vectors \(\overrightarrow{\mathbf{A}}\) and \(\overrightarrow{\mathbf{B}}\) with the cosine of the angle θ between the two vectors:
- https://phys.libretexts.org/Courses/Prince_Georges_Community_College/General_Physics_I%3A_Classical_Mechanics/35%3A_The_Cross_Product/35.02%3A_PropertiesA better way to remember both products in Eqs. (32.12) and (32.13) is: "The middle vector times the dot product of the two on the ends, minus the dot product of the two vectors straddling the parenthe...A better way to remember both products in Eqs. (32.12) and (32.13) is: "The middle vector times the dot product of the two on the ends, minus the dot product of the two vectors straddling the parenthesis times the remaining one.”
- https://phys.libretexts.org/Courses/Prince_Georges_Community_College/General_Physics_I%3A_Classical_Mechanics/10%3A_The_Dot_Product/10.03%3A_PropertiesFor example, dotting a vector \(\mathbf{A}\) with any of the cartesian unit vectors gives the projection of \(\mathbf{A}\) in that direction: In general, the projection of vector \(\mathbf{A}\) in the...For example, dotting a vector \(\mathbf{A}\) with any of the cartesian unit vectors gives the projection of \(\mathbf{A}\) in that direction: In general, the projection of vector \(\mathbf{A}\) in the direction of unit vector \(\hat{\mathbf{u}}\) is \(\mathbf{A} \cdot \hat{\mathbf{u}}\). From Eq. (7.2.6), it follows that \(\mathbf{A} \cdot \mathbf{A}=A_{x}^{2}+A_{y}^{2}+A_{z}^{2}=A^{2}\); so the magnitude of a vector \(\mathbf{A}\) is given in terms of the dot product by
- https://phys.libretexts.org/Workbench/PH_245_Textbook_V2/01%3A_Module_0_-_Mathematical_Foundations/1.02%3A_Objective_0.b./1.2.07%3A_Products_of_Vectors_(Part_1)One kind of vector multiplication is the scalar product, also known as the dot product, which results in a number (scalar). The scalar product has the distributive property and the commutative propert...One kind of vector multiplication is the scalar product, also known as the dot product, which results in a number (scalar). The scalar product has the distributive property and the commutative property, and is obtained by multiplying the magnitudes of the two vectors with the cosine of the angle between them. This type of vector multiplication is used to find angles between vectors and in the definitions of derived scalar physical quantities such as work or energy.
- https://phys.libretexts.org/Courses/Joliet_Junior_College/Physics_201_-_Fall_2019/Book%3A_Physics_(Boundless)/03%3A_Vectors/3.08%3A_Products_of_Vectors_(Part_1)One kind of vector multiplication is the scalar product, also known as the dot product, which results in a number (scalar). The scalar product has the distributive property and the commutative propert...One kind of vector multiplication is the scalar product, also known as the dot product, which results in a number (scalar). The scalar product has the distributive property and the commutative property, and is obtained by multiplying the magnitudes of the two vectors with the cosine of the angle between them. This type of vector multiplication is used to find angles between vectors and in the definitions of derived scalar physical quantities such as work or energy.
- https://phys.libretexts.org/Bookshelves/University_Physics/University_Physics_(OpenStax)/Book%3A_University_Physics_I_-_Mechanics_Sound_Oscillations_and_Waves_(OpenStax)/02%3A_Vectors/2.08%3A_Products_of_Vectors_(Part_1)One kind of vector multiplication is the scalar product, also known as the dot product, which results in a number (scalar). The scalar product has the distributive property and the commutative propert...One kind of vector multiplication is the scalar product, also known as the dot product, which results in a number (scalar). The scalar product has the distributive property and the commutative property, and is obtained by multiplying the magnitudes of the two vectors with the cosine of the angle between them. This type of vector multiplication is used to find angles between vectors and in the definitions of derived scalar physical quantities such as work or energy.
