“vector product”的概念、定义、翻译、参考文献-科学参考

vector product

Physics

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

$U \times V = (U_{2} V_{3} - U_{3} V_{2}) i + (U_{3} V_{1} - U_{1} V_{3}) j + (U_{1} V_{2} - U_{2} V_{1}) k$

It is itself a vector, perpendicular to both U and V, and of length UVsinθ‎, where U and V are the lengths of U and V, respectively, and θ‎ is the angle between them. Compare scalar product.

Vector product.

Mathematics

Let a and b be non-zero vectors, and let θ‎ be the angle between them (θ‎ in radians, with 0 ≤ θ‎ < π‎). The vector product a × b of a and b is defined as having magnitude |a∥b|sinθ‎, and (if non-zero) its direction is perpendicular to a and b such that a, b and a × b form a right-handed system. The notation a∧b is also used for a × b. If a = a₁i + a₂j + a₃k, b = b₁i + b₂j + b₃k, then a × b = (a₂b₃ − a₃b₂)i + (a₃b₁ − a₁b₃)j + (a₁b₂ − a₂b₁)k. This can be written, using 3 × 3 determinant notation, as

$a \times b = | \begin{matrix} i & j & k \\ a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \end{matrix} | .$

The following properties hold, for all vectors a, b, and c:
1. (i) b × a = −(a × b).
2. (ii) a × (kb + lc) = k(a × b) + l(a × c).
3. (iii) a × b is perpendicular to a and b.
4. (iv) If a and b are perpendicular unit vectors, then a × b is a unit vector.
In fact, these four properties determine the vector product up to a choice of sign. Unlike the scalar product, which can be defined in any dimension, a vector product with these properties only exists in 3 and 7 dimensions.

单词	vector product
释义	vector product Physics The product of two vectors U and V, with components U₁, U₂, U₃ and V₁, V₂, V₃, respectively, given by: $U \times V = (U_{2} V_{3} - U_{3} V_{2}) i + (U_{3} V_{1} - U_{1} V_{3}) j + (U_{1} V_{2} - U_{2} V_{1}) k$ It is itself a vector, perpendicular to both U and V, and of length UVsinθ‎, where U and V are the lengths of U and V, respectively, and θ‎ is the angle between them. Compare scalar product. Vector product. Mathematics Let a and b be non-zero vectors, and let θ‎ be the angle between them (θ‎ in radians, with 0 ≤ θ‎ < π‎). The vector product a × b of a and b is defined as having magnitude \|a∥b\|sinθ‎, and (if non-zero) its direction is perpendicular to a and b such that a, b and a × b form a right-handed system. The notation a∧b is also used for a × b. If a = a₁i + a₂j + a₃k, b = b₁i + b₂j + b₃k, then a × b = (a₂b₃ − a₃b₂)i + (a₃b₁ − a₁b₃)j + (a₁b₂ − a₂b₁)k. This can be written, using 3 × 3 determinant notation, as $a \times b = \| \begin{matrix} i & j & k \\ a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \end{matrix} \| .$ The following properties hold, for all vectors a, b, and c: (i) b × a = −(a × b). (ii) a × (kb + lc) = k(a × b) + l(a × c). (iii) a × b is perpendicular to a and b. (iv) If a and b are perpendicular unit vectors, then a × b is a unit vector. In fact, these four properties determine the vector product up to a choice of sign. Unlike the scalar product, which can be defined in any dimension, a vector product with these properties only exists in 3 and 7 dimensions.
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