One-Dimensional Displacement Vectors and Their Representations | 一维位移向量及其表示方法

📚 One-Dimensional Displacement Vectors and Their Representations | 一维位移向量及其表示方法

In physics and mathematics, a displacement vector describes a change in position. In one dimension, this vector is represented by a single signed number or a directed interval along a line. This article explains its definition, representations, operations, and applications.

在物理和数学中，位移向量描述位置的变化。在一维情形下，这个向量可以用一个带符号的数或直线上的有向区间来表示。本文将解释它的定义、表示方法、运算及应用。

1. What Is a One-Dimensional Displacement Vector? | 什么是一维位移向量？

A one-dimensional displacement vector is the change in position of an object along a straight line. It has both magnitude (size) and direction (positive or negative). It is written as Δx = x₂ − x₁, where x₁ is the initial position and x₂ is the final position.

一维位移向量是物体沿直线运动时的位置变化。它既有大小也有方向（正或负）。记作 Δx = x₂ − x₁，其中 x₁ 是初位置，x₂ 是末位置。

Δx = x₂ − x₁

If Δx is positive, the displacement is in the positive direction; if negative, it is in the negative direction.

若 Δx 为正，则位移沿正方向；若为负，则位移沿负方向。

2. Vector vs. Scalar: Displacement vs. Distance | 向量与标量：位移与路程

Displacement is a vector quantity; distance is a scalar quantity. Distance only measures how much ground is covered, while displacement measures net change in position including direction.

位移是向量，路程是标量。路程只度量经过路径的总长度，而位移度量位置的净变化并包含方向。

Distance is always non-negative and does not have direction. 路程始终非负，没有方向。
Displacement can be positive, negative, or zero. 位移可以为正、负或零。

Example: A person walks 5 m right and then 5 m left. Distance = 10 m; displacement = 0 m.

例：一个人先向右走 5 m，再向左走 5 m。路程 = 10 m；位移 = 0 m。

3. Representing a One-Dimensional Vector by a Signed Number | 用带符号的数表示一维向量

The simplest representation is a single real number with a sign. For example, +3 m means 3 metres in the positive direction; −3 m means 3 metres in the negative direction.

最简单的表示方法是带符号的实数。例如 +3 m 表示沿正方向 3 米；−3 m 表示沿负方向 3 米。

The magnitude of the vector is the absolute value: |Δx|. For −3 m, the magnitude is 3 m.

向量的大小是绝对值：|Δx|。对于 −3 m，其大小为 3 m。

magnitude = |Δx| = √(Δx)²

In one dimension, the sign encodes direction.

在一维中，符号编码了方向。

4. Vector Notation in One Dimension | 一维向量的符号记法

In mathematics, a one-dimensional vector may be written as a column vector or using an arrow. For example, the displacement from 2 m to 5 m is Δx = 3 m. In unit-vector form, it can be written as 3 i, where i is the unit vector in the positive x-direction.

在数学中，一维向量可以写成列向量或使用箭头。例如，从 2 m 到 5 m 的位移为 Δx = 3 m。用单位向量形式可写为 3 i，其中 i 是正 x 方向的单位向量。

Sometimes it is written as a row vector or just as a coordinate on a number line. The key is that one number is enough.

有时也可写成行向量或数轴上的坐标。关键在于一个数就足够了。

5. Position Vectors and Displacement | 位置向量与位移

A position vector gives the location of a point relative to an origin. In one dimension, the position vector of a point is simply its coordinate x. The displacement from position A to position B is the difference of their position vectors.

位置向量表示某点相对于原点的位置。在一维中，点的位置向量就是其坐标 x。从位置 A 到位置 B 的位移等于它们的位置向量之差。

Let the position vectors of two points be x_A and x_B. Then the displacement vector from A to B is:

设两点的位置向量分别为 x_A 和 x_B。那么从 A 到 B 的位移向量为：

Δx = x_B − x_A

This shows that displacement depends only on initial and final positions, not on the path taken.

这说明位移只取决于初末位置，而与运动路径无关。

6. The Zero Vector and Negative Vector | 零向量与负向量

The zero vector in one dimension is 0. It represents no displacement or a return to the starting point. Its magnitude is zero and its direction is undefined.

一维中的零向量是 0。它表示没有位移或回到起点。其大小为 0，方向未定义。

The negative of a vector v is −v. It has the same magnitude but opposite direction. For example, if v = +5 m, then −v = −5 m.

向量 v 的负向量是 −v。它的大小相同但方向相反。例如，若 v = +5 m，则 −v = −5 m。

In displacement terms: if you move from A to B, the displacement is v; moving back from B to A gives the displacement −v.

用位移的语言来说：若从 A 到 B 的位移为 v，则从 B 回到 A 的位移为 −v。

7. Adding and Subtracting One-Dimensional Vectors | 一维向量的加法与减法

In one dimension, vector addition is ordinary algebraic addition with signs. If you first move +4 m and then +2 m, the total displacement is +6 m. If you move +4 m and then −2 m, the total is +2 m.

在一维中，向量加法就是带符号的普通代数加法。若先移动 +4 m，再移动 +2 m，总位移为 +6 m。若先移动 +4 m，再移动 −2 m，则总位移为 +2 m。

v_total = v₁ + v₂

Subtraction of vectors is the same as adding the negative: v₁ − v₂ = v₁ + (−v₂).

向量减法等同于加上负向量：v₁ − v₂ = v₁ + (−v₂)。

8. Scalar Multiplication of a One-Dimensional Vector | 一维向量的数乘

Multiplying a vector by a scalar k changes its magnitude by a factor |k| and reverses direction if k is negative. In one dimension, this is simply multiplication of the signed number by k.

向量与标量 k 相乘，其大小变为原来的 |k| 倍；若 k 为负，则方向反向。在一维中，就是带符号的数乘以 k。

For example, 3 × (2 m) = 6 m; (−2) × (3 m) = −6 m. The magnitude becomes 6 m in both cases, but directions are opposite.

例如，3 × (2 m) = 6 m；(−2) × (3 m) = −6 m。两种情况下大小都是 6 m，但方向相反。

9. Displacement-Time Graphs and Velocity | 位移-时间图像与速度

The one-dimensional displacement vector is often studied with a displacement–time graph. The gradient of the graph gives the velocity, which is also a one-dimensional vector.

一维位移向量经常通过位移-时间图像来研究。图像的斜率给出速度，速度也是一维向量。

If the displacement changes from x₁ to x₂ over time t₂ − t₁, the average velocity is:

若位移在时间 t₂ − t₁ 内从 x₁ 变化到 x₂，则平均速度为：

v_avg = (x₂ − x₁) / (t₂ − t₁)

The sign of v_avg shows the direction of motion.

v_avg 的符号表示运动方向。

10. Representing Vectors on a Number Line | 在数轴上表示向量

One-dimensional vectors can be drawn as directed line segments on a number line. An arrow from the initial point to the final point shows both magnitude and direction.

一维向量可以画成数轴上的有向线段。从起点指向终点的箭头同时显示大小和方向。

For example, a displacement from 2 m to 7 m is drawn as an arrow of length 5 m pointing to the right. The coordinate difference is +5 m.

例如，从 2 m 到 7 m 的位移画成长度为 5 m 且指向右方的箭头。坐标差为 +5 m。

This visual representation clarifies why the sign convention matters.

这种图示清晰地说明了为什么符号约定很重要。

11. Applications in Kinematics and Integration | 在运动学与积分中的应用

In kinematics, total displacement can be found by integrating velocity over time: Δx = ∫ v(t) dt. This is a one-dimensional vector integral.

在运动学中，总位移可以通过对速度随时间积分求得：Δx = ∫ v(t) dt。这是一个一维向量积分。

If v(t) is positive for part of the motion and negative for another, the integral automatically combines directional changes.

如果 v(t) 在运动的一部分为正，另一部分为负，积分会自动合并方向变化。

Similarly, the area under a velocity–time graph represents displacement, with areas below the time axis counted as negative.

类似地，速度-时间图像下的面积表示位移，时间轴下方的面积计为负值。

12. Common Mistakes and Exam Tips | 常见错误与考试提示

Students often confuse displacement with distance. Remember that displacement is the net change in position, not the total path length.

学生常常混淆位移与路程。记住位移是位置的净变化，而不是路径总长度。

Always define a positive direction at the start of a problem and be consistent with signs throughout.

解题开始时务必规定正方向，并全程保持符号一致。

When calculating displacement, use final position minus initial position, never the reverse.

计算位移时，要用末位置减初位置，切勿颠倒。

Finally, for IB exams, show the vector nature explicitly: write Δx = x₂ − x₁ and include units and direction.

最后，在 IB 考试中，要明确展示向量性质：写出 Δx = x₂ − x₁，并带上单位与方向。

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One-Dimensional Displacement Vectors and Their Representations | 一维位移向量及其表示方法