“vector product”的概念、定义、习题解题方法及翻译-数学辞典

单词

vector product

释义

vector product

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

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.

随便看

数学辞典收录了4151条数学词条，基本涵盖了常用数学知识及数学英语单词词组的翻译及用法，是数学学习的有利工具。