Basic Concepts of Vectors in a Plane | 平面向量的基本概念

📚 Basic Concepts of Vectors in a Plane | 平面向量的基本概念

Vectors are one of the most fundamental tools in IB Mathematics, providing a bridge between geometry and algebra. In this article, we will explore the essential definitions and operations that form the foundation of vector geometry in the plane.

向量是IB数学中最基础的工具之一，它连接了几何与代数。在本文中，我们将探索平面向量几何中构成基础的核心定义与运算。

1. What Is a Vector? | 什么是向量？

A vector is a quantity that has both magnitude (size) and direction. For example, displacement, velocity, and force are all vector quantities.

向量是既有大小又有方向的量。例如，位移、速度和力都是向量。

In contrast, a scalar only has magnitude, such as temperature, mass, or speed. A vector is usually represented by a directed line segment, with an arrow indicating its direction.

相比之下，标量只有大小，例如温度、质量或速率。向量通常用有向线段表示，箭头指示方向。

\(\overrightarrow{AB}\) denotes the vector from point A to point B.

\(\overrightarrow{AB}\) 表示从点 A 到点 B 的向量。

\(\vec{v} = \begin{pmatrix} x \\ y \end{pmatrix}\) in component form.

\(\vec{v} = \begin{pmatrix} x \\ y \end{pmatrix}\) 为分量形式。

2. Vector vs Scalar | 向量与标量的区别

A scalar is described entirely by a single real number, while a vector requires both a number and a direction.

标量可以用一个实数完全描述，而向量则需要数值和方向两个要素。

Scalar examples: mass 5 kg, temperature 20 °C, speed 60 km/h.
向量示例：质量 5 kg、温度 20 °C、速率 60 km/h。
Vector examples: displacement 5 km north, velocity 60 km/h east, force 10 N downward.
向量示例：位移 5 km 向北、速度 60 km/h 向东、力 10 N 向下。

3. Equal Vectors and Negative Vectors | 相等向量与负向量

Two vectors are equal if they have the same magnitude and the same direction, regardless of their starting point.

两个向量相等当且仅当它们的大小相同且方向相同，与起点无关。

The negative of a vector \(\vec{v}\), written as \(-\vec{v}\), has the same magnitude but exactly the opposite direction.

向量 \(\vec{v}\) 的负向量记作 \(-\vec{v}\)，它的大小相同，但方向完全相反。

If \(\vec{AB} = \vec{CD}\), then the two directed segments are parallel, equal in length, and point in the same direction.

若 \(\vec{AB} = \vec{CD}\)，则两条有向线段平行、等长且同向。

4. Magnitude of a Vector | 向量的模

The magnitude (or length) of a vector \(\vec{v} = \begin{pmatrix} x \\ y \end{pmatrix}\) is given by the distance formula.

向量 \(\vec{v} = \begin{pmatrix} x \\ y \end{pmatrix}\) 的模（即长度）由距离公式给出。

\(|\vec{v}| = \sqrt{x^2 + y^2}\)

For example, if \(\vec{v} = \begin{pmatrix} 3 \\ -4 \end{pmatrix}\), then \(|\vec{v}| = \sqrt{3^2 + (-4)^2} = 5\).

例如，若 \(\vec{v} = \begin{pmatrix} 3 \\ -4 \end{pmatrix}\)，则 \(|\vec{v}| = \sqrt{3^2 + (-4)^2} = 5\)。

5. Unit Vectors | 单位向量

A unit vector is a vector with magnitude exactly 1. It is often used to indicate direction only.

单位向量是模等于 1 的向量，通常仅用于表示方向。

To find a unit vector in the direction of \(\vec{v}\), divide \(\vec{v}\) by its magnitude:

求与 \(\vec{v}\) 同方向的单位向量，可将 \(\vec{v}\) 除以它的模：

\(\hat{v} = \frac{\vec{v}}{|\vec{v}|}\)

If \(\vec{v} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}\), then \(|\vec{v}| = 5\), so \(\hat{v} = \begin{pmatrix} \frac{3}{5} \\ \frac{4}{5} \end{pmatrix}\).

若 \(\vec{v} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}\)，则 \(|\vec{v}| = 5\)，因此 \(\hat{v} = \begin{pmatrix} \frac{3}{5} \\ \frac{4}{5} \end{pmatrix}\)。

6. Position Vectors and Free Vectors | 位置向量与自由向量

A position vector is a vector that starts from the origin O. It describes the location of a point relative to O.

位置向量是以原点 O 为起点的向量，它描述了点相对于原点 O 的位置。

For a point \(A(x, y)\), its position vector is written as \(\vec{OA} = \begin{pmatrix} x \\ y \end{pmatrix}\).

对于点 \(A(x, y)\)，其位置向量记为 \(\vec{OA} = \begin{pmatrix} x \\ y \end{pmatrix}\)。

A free vector can be translated anywhere in the plane without changing its value, because only its magnitude and direction matter.

自由向量可以在平面内任意平移而不改变其值，因为只有大小和方向起作用。

7. Vector Addition | 向量的加法

Vector addition can be performed using algebra, the triangle rule, or the parallelogram rule.

向量加法可以用代数法、三角形法则或平行四边形法则进行。

Algebraically, if \(\vec{a} = \begin{pmatrix} a_1 \\ a_2 \end{pmatrix}\) and \(\vec{b} = \begin{pmatrix} b_1 \\ b_2 \end{pmatrix}\), then

代数上，若 \(\vec{a} = \begin{pmatrix} a_1 \\ a_2 \end{pmatrix}\)，\(\vec{b} = \begin{pmatrix} b_1 \\ b_2 \end{pmatrix}\)，则

\(\vec{a} + \vec{b} = \begin{pmatrix} a_1 + b_1 \\ a_2 + b_2 \end{pmatrix}\)

Triangle rule: place the tail of \(\vec{b}\) at the head of \(\vec{a}\); the sum joins the tail of \(\vec{a}\) to the head of \(\vec{b}\).

三角形法则：将 \(\vec{b}\) 的起点放在 \(\vec{a}\) 的终点，和向量从 \(\vec{a}\) 的起点指向 \(\vec{b}\) 的终点。

Parallelogram rule: when both vectors start from the same point, the diagonal of the parallelogram gives the sum.

平行四边形法则：两个向量从同一起点出发，平行四边形的对角线即为和向量。

8. Scalar Multiplication | 数乘向量

Multiplying a vector by a scalar \(k\) changes its length but not its direction (unless \(k < 0\), which reverses the direction).

向量乘以标量 \(k\) 会改变其长度但不改变方向（若 \(k < 0\)，则方向反转）。

If \(\vec{v} = \begin{pmatrix} x \\ y \end{pmatrix}\), then

若 \(\vec{v} = \begin{pmatrix} x \\ y \end{pmatrix}\)，则

\(k\vec{v} = \begin{pmatrix} kx \\ ky \end{pmatrix}\)

Also, \(|k\vec{v}| = |k| \, |\vec{v}|\).

同时，\(|k\vec{v}| = |k| \, |\vec{v}|\)。

9. Linear Combinations and Basis Vectors | 线性组合与基本向量

In two dimensions, any vector can be expressed as a linear combination of two independent basis vectors. The standard basis vectors are

在二维平面中，任何向量都可以表示为两个不共线基向量的线性组合。标准基向量为

\(\vec{i} = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \quad \vec{j} = \begin{pmatrix} 0 \\ 1 \end{pmatrix}\)

Thus any vector \(\vec{v} = \begin{pmatrix} x \\ y \end{pmatrix}\) can be written as \(\vec{v} = x\vec{i} + y\vec{j}\).

因此任何向量 \(\vec{v} = \begin{pmatrix} x \\ y \end{pmatrix}\) 都可写成 \(\vec{v} = x\vec{i} + y\vec{j}\)。

This representation is extremely useful in solving problems involving geometry and motion.

这种表示在解决几何和运动问题时极为有用。

10. Parallel Vectors and Collinearity | 平行向量与共线

Two nonzero vectors are parallel if one is a scalar multiple of the other. That is, \(\vec{a} = k\vec{b}\) for some scalar \(k\).

两个非零向量平行当且仅当其中一个可以写成另一个的标量倍，即存在标量 \(k\) 使 \(\vec{a} = k\vec{b}\)。

If \(\vec{a} = \begin{pmatrix} a_1 \\ a_2 \end{pmatrix}\) and \(\vec{b} = \begin{pmatrix} b_1 \\ b_2 \end{pmatrix}\) are parallel, then

若 \(\vec{a} = \begin{pmatrix} a_1 \\ a_2 \end{pmatrix}\) 和 \(\vec{b} = \begin{pmatrix} b_1 \\ b_2 \end{pmatrix}\) 平行，则

\(\frac{a_1}{b_1} = \frac{a_2}{b_2}\) (provided denominators are nonzero)

\(\frac{a_1}{b_1} = \frac{a_2}{b_2}\)（分母不为零时）

Collinearity: three points A, B, C are collinear if \(\vec{AB}\) and \(\vec{AC}\) are parallel, i.e. \(\vec{AB} = t\vec{AC}\) for some scalar \(t\).

共线：三点 A、B、C 共线当且仅当 \(\vec{AB}\) 与 \(\vec{AC}\) 平行，即存在标量 \(t\) 使 \(\vec{AB} = t\vec{AC}\)。

11. The Dot Product (Scalar Product) | 向量的点积（数量积）

The dot product of two vectors gives a scalar and is a key tool for finding angles between vectors.

两个向量的点积得到一个标量，它是求向量夹角的重要工具。

Algebraically, if \(\vec{a} = \begin{pmatrix} a_1 \\ a_2 \end{pmatrix}\) and \(\vec{b} = \begin{pmatrix} b_1 \\ b_2 \end{pmatrix}\), then

代数上，若 \(\vec{a} = \begin{pmatrix} a_1 \\ a_2 \end{pmatrix}\)，\(\vec{b} = \begin{pmatrix} b_1 \\ b_2 \end{pmatrix}\)，则

\(\vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2\)

Geometrically, \(\vec{a} \cdot \vec{b} = |\vec{a}| \, |\vec{b}| \cos \theta\), where \(\theta\) is the angle between the vectors.

几何上，\(\vec{a} \cdot \vec{b} = |\vec{a}| \, |\vec{b}| \cos \theta\)，其中 \(\theta\) 是两个向量的夹角。

If two vectors are perpendicular, their dot product is zero: \(\vec{a} \cdot \vec{b} = 0\).

若两个向量垂直，则它们的点积为零：\(\vec{a} \cdot \vec{b} = 0\)。

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Basic Concepts of Vectors in a Plane | 平面向量的基本概念