The dot product is the most common way to define an inner product between elements of (-dimensional vectors). Note that some texts use the symbol to denote the dot product between and , preserving the inner-product notation. The dot product is one of three common types of multiplication compatible with vectors; the other being the cross product and scalar multiplication, the latter belonging to the vector space nature of .
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| - The dot product is the most common way to define an inner product between elements of (-dimensional vectors). Note that some texts use the symbol to denote the dot product between and , preserving the inner-product notation. The dot product is one of three common types of multiplication compatible with vectors; the other being the cross product and scalar multiplication, the latter belonging to the vector space nature of .
- The dot product is a vector operation, which, along with the cross product and scalar multiplication, comprise the three types of "multiplication" operations that can be performed on vectors. Vector division is not defined. The fundamental definition of a dot product is the product of the scalar magnitudes (lengths) of each vector and the cosine of the angle in between them. This definition can be equivalently written as a scalar projection, or component, of vector u onto vector v, times the magnitude of vector v:
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* The dot product is:
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| - The dot product is the most common way to define an inner product between elements of (-dimensional vectors). Note that some texts use the symbol to denote the dot product between and , preserving the inner-product notation. The dot product is one of three common types of multiplication compatible with vectors; the other being the cross product and scalar multiplication, the latter belonging to the vector space nature of .
- The dot product is a vector operation, which, along with the cross product and scalar multiplication, comprise the three types of "multiplication" operations that can be performed on vectors. Vector division is not defined. The fundamental definition of a dot product is the product of the scalar magnitudes (lengths) of each vector and the cosine of the angle in between them. This definition can be equivalently written as a scalar projection, or component, of vector u onto vector v, times the magnitude of vector v: By consequence of the definition, a dot product is the sum of the products of corresponding elements within the vector. This is often regarded as an alternative definition. Given the two (3-dimensional) vectors u and v:
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* The dot product is:
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