V2EX scalar product

Scalar Product

定义 Definition

scalar product(数学)指两个向量的乘积结果是一个标量(单个数)的运算,最常见的是点积(dot product)
\[ \mathbf{a}\cdot\mathbf{b}=|a||b|\cos\theta \] 在欧几里得空间中,它常用来衡量两个向量的“对齐程度”,并用于计算投影、夹角、正交性等。(在某些语境里也可泛指更一般的“内积”,但最常见用法是点积。)

发音 Pronunciation (IPA)

/skelr prdkt/

例句 Examples

The scalar product of two perpendicular vectors is zero.
两条互相垂直的向量的标量积为零。

In physics, we often use the scalar product to compute work as the dot product of force and displacement.
在物理中,我们常用标量积来计算功,即力与位移的点积。

词源 Etymology

scalar 来自拉丁语 scalaris,与“梯子(scala)”相关,引申为“分等级的、标量的”;在数学里 scalar 指“单个数值”。product 源自拉丁语 productus(“产生、引出”),在数学中指“乘积”。合起来 scalar produc 就是“产生一个标量结果的乘积”,在向量运算中通常指点积。

相关词 Related Words

文学与著作中的用例 Literary & Notable Works

  • Gilbert Strang, Introduction to Linear Algebra:在向量与几何章节中用标量积(点积)解释投影与正交。
  • Sheldon Axler, Linear Algebra Done Right:在讨论内积空间时涉及标量积/内积的概念与性质。
  • David C. Lay, Linear Algebra and Its Applications:以标量积为基础引入夹角、长度与正交性等应用。
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