Vector Geometry | 向量几何

📚 Vector Geometry | 向量几何

Vectors are fundamental in understanding geometry and motion, providing a way to describe quantities that have both magnitude and direction. In IGCSE Mathematics, vector geometry combines algebraic techniques with geometric insight, enabling you to prove relationships, find unknown points, and solve real‑world problems with precision.

向量是理解几何与运动的基础，可用于描述既有大小又有方向的量。在 IGCSE 数学中，向量几何将代数技巧与几何直观结合起来，帮助你证明关系、求解未知点，并精确地解决实际问题。

1. What is a Vector? | 什么是向量？

A vector is a quantity that has both magnitude (size) and direction. This contrasts with a scalar, which has magnitude only. Examples of vectors include displacement, velocity, acceleration and force. Scalars include distance, speed, mass and temperature. In a diagram, a vector is drawn as a directed line segment: the length represents its magnitude, and the arrowhead shows its direction.

向量是既有大小（模）又有方向的量。这与仅有大小的标量形成对比。向量的例子有位移、速度、加速度和力。标量包括路程、速率、质量和温度。在图形中，向量表示为有向线段：长度代表大小，箭头表示方向。

Vectors are denoted by bold letters such as a or by letters with an arrow, like a⃗. When handwritten, it is common to underline the letter, e.g., a. The vector from point A to point B is written as AB. Two vectors are equal if they have the same magnitude and the same direction, regardless of their starting point.

向量常用粗体字母a或带箭头的字母（如 a⃗）表示。手写时通常加下划线，如 a 。从点 A 指向点 B 的向量记为 AB。两个向量只要大小相等且方向相同，无论起点在哪里，它们就是相等的。

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2. Representing Vectors | 向量的表示方法

In the Cartesian plane, a vector can be expressed as a column vector or in terms of the unit vectors i and j. The unit vector i points in the positive x‑direction with magnitude 1, while j points in the positive y‑direction with magnitude 1. A vector that moves x units horizontally and y units vertically is written as xi + yj. For example, a = 3i − 2j.

在笛卡儿平面中，向量可表示为列向量或用单位向量 i 和 j 表示。单位向量 i 指向 x 轴正方向，大小为 1；j 指向 y 轴正方向，大小也为 1。水平移动 x 单位、竖直移动 y 单位的向量记作 xi + yj。例如，a = 3i − 2j。

Column vector notation writes the components vertically, with the x‑component on top. While the vertical layout is standard in exams, it is often typed as (3, −2) for convenience. The table below shows some vectors and their equivalent forms.

列向量表示法将分量上下排列，x 分量在上。考试中通常使用竖直书写，但在文字中常简写为 (3, −2)。下表给出几个向量及其等价形式。

Vector	i, j form	Column vector (concept)
p	5i + 2j	(5, 2) with 5 above 2
q	−i + 4j	(−1, 4) vertically
r	−3i − j	(−3, −1) vertically

These representations are interchangeable. When adding, subtracting or multiplying by a scalar, it is usually easier to work with the i, j form or column components directly.

这些表示形式可以互换。在进行加法、减法或标量乘法时，通常直接使用 i、j 形式或分量形式更为简便。

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3. Magnitude of a Vector | 向量的大小（模）

The magnitude (or modulus) of a vector a = xi + yj is its length. It is found using Pythagoras' theorem and is denoted by |a|.

向量 a = xi + yj 的大小（或模）就是它的长度。可通过勾股定理计算，记作 |a|。

|a| = √x² + y²

For example, if v = 3i − 4j, then |v| = √(3² + (−4)²) = √9 + 16 = √25 = 5. A vector of magnitude 1 is a unit vector. To find a unit vector in the direction of a, divide the vector by its magnitude: â = a / |a|.

例如，若 v = 3i − 4j，则 |v| = √(3² + (−4)²) = √9 + 16 = √25 = 5。大小为 1 的向量称为单位向量。要得到与 a 同方向的单位向量，只需将向量除以其大小：â = a / |a|。

The magnitude of the vector AB gives the distance between points A and B. This is particularly useful in coordinate geometry problems.

向量 AB 的大小就是 A、B 两点间的距离。这在坐标几何问题中特别有用。

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4. Vector Addition and Subtraction | 向量的加法和减法

Vectors can be added or subtracted by combining the i and j components separately. If a = x₁i + y₁j and b = x₂i + y₂j, then:

向量相加（减）只需将 i 分量和 j 分量分别相加减。设 a = x₁i + y₁j，b = x₂i + y₂j，则：

a + b = (x₁ + x₂)i + (y₁ + y₂)j

a − b = (x₁ − x₂)i + (y₁ − y₂)j

Geometrically, vector addition can be visualised using the Triangle Law or the Parallelogram Law. To add a and b, place the tail of b at the head of a; the sum is the vector from the start of a to the end of b. Subtraction a − b is equivalent to adding a and −b.

几何上，向量加法可通过三角形法则或平行四边形法则可视化。将 b 的起点放在 a 的终点上，和向量就是从 a 的起点指向 b 的终点的向量。减法 a − b 相当于 a 加上 −b。

The vector connecting two points AB can be found using subtraction of position vectors (discussed in Section 6). The zero vector 0 (0i + 0j) is the identity for addition.

连接两点的向量 AB 可通过位置向量相减得出（见第 6 节）。零向量 0（0i + 0j）是加法的单位元。

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5. Scalar Multiplication | 向量的标量乘法

Multiplying a vector by a scalar (a real number) changes the magnitude and possibly the direction. If a = xi + yj and k is a scalar, then ka = (kx)i + (ky)j. The magnitude becomes |k| |a|.

向量乘以一个标量（实数）会改变其大小，并可能改变方向。若 a = xi + yj，k 为标量，则 ka = (kx)i + (ky)j。大小变为 |k| |a|。

• If k > 0, ka is parallel to a and in the same direction.
• 若 k > 0，ka 与 a 平行且同向。
• If k < 0, ka is parallel to a but in the opposite direction.
• 若 k < 0，ka 与 a 平行但反向。
• If k = 0, we obtain the zero vector.
• 当 k = 0 时，得到零向量。

Scalar multiplication is essential when working with parallel vectors and when expressing one vector as a multiple of another. For instance, −2a is twice as long as a and points in the opposite direction.

标量乘法在处理平行向量以及将一个向量表示为另一个向量的倍数时至关重要。例如，−2a 的长度是 a 的两倍，且方向相反。

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6. Position Vectors and the Vector Between Two Points | 位置向量与两点间的向量

A position vector locates a point relative to a fixed origin O. If point P has coordinates (x, y), its position vector is p = OP = xi + yj. The vector from point A to point B can be expressed in terms of position vectors:

位置向量用于确定点相对于固定原点 O 的位置。若点 P 的坐标为 (x, y)，则其位置向量为 p = OP = xi + yj。从点 A 到点 B 的向量可用位置向量表示为：

AB = b − a

Here a and b are the position vectors of A and B. This formula is one of the most important in vector geometry. It allows us to describe any segment using known position vectors.

其中 a 和 b 分别是 A 和 B 的位置向量。这是向量几何中最重要的公式之一，使我们能用已知位置向量描述任意线段。

For example, if A is (2, 5) and B is (8, 1), then a = 2i + 5j, b = 8i + j, and AB = b − a = (8−2)i + (1−5)j = 6i − 4j. The midpoint M of AB would then be m = (a + b)/2.

例如，设 A(2,5)，B(8,1)，则 a = 2i + 5j，b = 8i + j，故 AB = b − a = (8−2)i + (1−5)j = 6i − 4j。此时 AB 的中点 M 的位置向量为 m = (a + b)/2。

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7. Parallel and Collinear Vectors | 平行向量与共线向量

Two vectors a and b are parallel if one is a scalar multiple of the other, i.e., a = kb for some scalar k ≠ 0. This implies they have the same or opposite direction. To check whether vectors are parallel, examine whether their components are in the same ratio.

两个向量 a 和 b 平行，当且仅当其中一个可以写成另一个的标量倍，即存在非零标量 k 使 a = kb。这意味着它们方向相同或相反。判断向量是否平行，可检查对应分量是否成相同比例。

Points A, B and C are collinear if they lie on the same straight line. This can be proved using vectors: if AB and BC are parallel (i.e., AB = k BC), and they share the point B, then A, B and C must be collinear. Often we express the vectors in terms of the same two base vectors and compare their coefficients.

若点 A、B、C 在同一直线上，则它们共线。这可用向量证明：若 AB 与 BC 平行（即 AB = k BC），且它们有公共点 B，则 A、B、C 必共线。我们通常将向量用同一组基向量表示，并比较系数。

For example, if AB = 2i +

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