Multiplication of a Vector by a Scalar | 向量与标量的乘法

📚 Multiplication of a Vector by a Scalar | 向量与标量的乘法

In IGCSE Edexcel Mathematics, vectors are fundamental tools for describing quantities that possess both magnitude and direction. One of the most important operations performed on vectors is multiplication by a scalar. This operation stretches, shrinks, or reverses a vector while keeping its line of action fixed, and it forms the basis for many more advanced topics in mechanics and coordinate geometry.

在爱德思 IGCSE 数学中，向量是描述既有大小又有方向的量的基本工具。对向量最重要的运算之一就是标量乘法。这一运算会拉伸、缩小或反转向量，同时保持其作用线不变，也是力学与坐标几何中许多进阶课题的基础。

1. Scalars and Vectors | 标量与向量

A scalar is a quantity that has only magnitude. Examples include mass (5 kg), temperature (20°C), and distance (10 m). In pure mathematics, scalars are simply real numbers such as 2, -7, ½, and √3.

标量是只有大小的量。例子包括质量（5 千克）、温度（20°C）和距离（10 米）。在纯数学中，标量就是实数，例如 2、-7、½ 和 √3。

A vector is a quantity that has both magnitude and direction. Displacement, velocity, and force are classic examples. A vector is commonly drawn as an arrow: the arrow length represents the magnitude, and the arrowhead indicates the direction of travel.

向量是同时具有大小和方向的量。位移、速度和力都是典型的例子。向量通常用箭头表示：箭头的长度代表大小，箭头指向代表方向。

Vectors can be expressed in several forms: column vectors, row vectors, or in terms of the unit vectors i and j. For example, the vector (3, 4) represents a displacement of 3 units in the x-direction and 4 units in the y-direction.

向量可以用多种形式表示：列向量、行向量，或借助单位向量 i 和 j。例如，向量 (3, 4) 表示在 x 方向位移 3 个单位、在 y 方向位移 4 个单位。

2. Definition of Scalar Multiplication | 标量乘法的定义

Multiplying a vector by a scalar k means multiplying each component of the vector by k. If v = (x, y), then k v = (k x, k y).

将向量乘以标量 k，意味着把向量的每一个分量都乘以 k。若 v = (x, y)，则 k v = (k x, k y)。

For a vector v = (x, y) and a scalar k: k v = (k x, k y)

The result is a new vector parallel to the original vector v, or anti-parallel when k is negative. The scalar multiplication is applied independently to the horizontal and vertical components.

结果得到一个新的向量，与原来的向量 v 平行；当 k 为负数时，与 v 反向平行。标量乘法分别独立地应用于水平分量和垂直分量。

For example, if v = (2, 3) and k = 4, then 4v = (8, 12). Every component of v has been multiplied by 4.

例如，若 v = (2, 3) 且 k = 4，则 4v = (8, 12)。v 的每一个分量都乘以了 4。

3. Geometric Interpretation | 几何解释

Geometrically, multiplying a vector v by a scalar k changes the length of the vector by a factor of |k|. When |k| > 1, the vector stretches; when 0 < |k| < 1, the vector shrinks.

从几何上看，将向量 v 乘以标量 k，会使向量的长度变为原来的 |k| 倍。当 |k| > 1 时，向量被拉伸；当 0 < |k| < 1 时，向量被压缩。

If v is drawn from the origin to point P, then 2v extends from the origin to a point twice as far along the same straight line. Similarly, ½v ends halfway along the same path.

如果 v 从原点画到点 P，那么 2v 从原点出发，在同一条直线上延伸到

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