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Mathematics · 术语表

什么是 Scalar product？

又称： norm

Definition 16.1 High School Mathematics · 第 16 章 — The Scalar Product in the Plane

The scalar product of two vectors $\vec u$ and $\vec v$ is the number

\vec u \cdot \vec v = \tfrac12\left(\norm{\vec u + \vec v}^2 - \norm{\vec u}^2 - \norm{\vec v}^2\right),

where $\norm{\vec u}$ denotes the length (norm) of $\vec u$ .

示例

Example 16.3

$\vec u\,(3, 1)$ and $\vec v\,(2, -4)$ : $\vec u \cdot \vec v = 6 - 4 = 2$ . The norms are $\norm{\vec u} = \sqrt{10}$ and $\norm{\vec v} = \sqrt{20}$ , so $\cos\theta = \frac{2}{\sqrt{10}\sqrt{20}} = \frac{2}{10\sqrt2} = \frac{\sqrt2}{10}$ : the angle is a bit less than $82^\circ$ .

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Definition 31.6 High School Mathematics · 第 31 章 — Vectors, Lines and Planes in Space

The scalar product of $\vec u$ and $\vec v$ is

\vec u \cdot \vec v = \tfrac12\left(\norm{\vec u + \vec v}^2 - \norm{\vec u}^2 - \norm{\vec v}^2\right),

where $\norm{\vec u}$ is the length of $\vec u$ . If both vectors are nonzero, $\vec u \cdot \vec v = \norm{\vec u}\,\norm{\vec v}\cos\theta$ where $\theta$ is the angle between them; and in an orthonormal coordinate system,

\vec u \cdot \vec v = xx' + yy' + zz' .

Two vectors are orthogonal if their scalar product is $0$ .

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