“scalar product”的概念、定义、翻译、参考文献-科学参考

scalar product

Physics

The product of two vectors U and V, with components U₁, U₂, U₃ and V₁, V₂, V₃, respectively, given by:

$U . V = U_{1} V_{1} + U_{2} V_{2} U_{3} V_{3}$

It can also be written as UVcosθ‎, where U and V are the lengths of U and V, respectively, and θ‎ is the angle between them. Compare vector product.

Scalar product.

Mathematics

The Euclidean inner product. For vectors u and v in ℝ³, the scalar product is defined by

$u \cdot v = u_{1} v_{1} + u_{2} v_{2} + u_{3} v_{3},$

and this generalizes naturally to higher dimensions.

The scalar product has the following properties:
1. (i) u·v = v·u.
2. (ii) For non-zero vectors u and v, u·v = 0 if and only if u is perpendicular to v.
3. (iii) u·u = |u|²; the scalar product u·u may be written u².
4. (iv) u·(v+w)=u·v+u·w, the distributive law.
5. (v) u·(kv) = k(u·v).
The length |u| of a vector can be defined in terms of the scalar product from (iii). Further, the angle θ‎ between u and v can be defined by

$u \cdot v = | u | | v | cos θ .$

This uniquely defines an angle θ‎ in the range 0 ≤ θ‎ ≤ π‎ by the Cauchy-Schwarz inequality.

单词	scalar product
释义	scalar product Physics The product of two vectors U and V, with components U₁, U₂, U₃ and V₁, V₂, V₃, respectively, given by: $U . V = U_{1} V_{1} + U_{2} V_{2} U_{3} V_{3}$ It can also be written as UVcosθ‎, where U and V are the lengths of U and V, respectively, and θ‎ is the angle between them. Compare vector product. Scalar product. Mathematics The Euclidean inner product. For vectors u and v in ℝ³, the scalar product is defined by $u \cdot v = u_{1} v_{1} + u_{2} v_{2} + u_{3} v_{3},$ and this generalizes naturally to higher dimensions. The scalar product has the following properties: (i) u·v = v·u. (ii) For non-zero vectors u and v, u·v = 0 if and only if u is perpendicular to v. (iii) u·u = \|u\|²; the scalar product u·u may be written u². (iv) u·(v+w)=u·v+u·w, the distributive law. (v) u·(kv) = k(u·v). The length \|u\| of a vector can be defined in terms of the scalar product from (iii). Further, the angle θ‎ between u and v can be defined by $u \cdot v = \| u \| \| v \| cos θ .$ This uniquely defines an angle θ‎ in the range 0 ≤ θ‎ ≤ π‎ by the Cauchy-Schwarz inequality.
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