IB Math: Essential Basics of Vectors | 向量入门必备基础知识梳理

📚 IB Math: Essential Basics of Vectors | 向量入门必备基础知识梳理

Vectors are one of the foundational topics in IB Mathematics. They appear throughout the syllabus in geometry, kinematics, and even in calculus-based applications. Whether you are studying Analysis and Approaches or Applications and Interpretation, a clear understanding of vector basics is essential for solving problems confidently and avoiding careless mistakes.

向量是IB数学中的基础课题之一，贯穿几何、运动学以及微积分相关应用。无论你学习的是分析与方法（AA）还是应用与解释（AI），清晰理解向量基础对于自信解题、避免粗心错误都至关重要。本文为你梳理向量入门必须掌握的核心知识。

1. What is a Vector? | 什么是向量

A vector is a quantity that has both magnitude (size) and direction. Examples include displacement, velocity, and force. A scalar, by contrast, has only magnitude, such as mass, speed, or temperature.

向量是既有大小（模长）又有方向的量，例如位移、速度和力。而标量只有大小，没有方向，例如质量、速率和温度。

We usually denote a vector by a bold lowercase letter such as v, or by writing an arrow over the letter. In printed texts IB often uses bold type, while in handwritten work you should draw an arrow: v⃗. The magnitude of vector v is written as |v|.

我们通常用粗体小写字母表示向量，例如 v，或在字母上方加箭头表示。IB 教材中通常使用粗体，手写时则建议加箭头：v⃗。向量 v 的模长记作 |v|。

2. Component Form and Magnitude | 向量的分量表示与模长

In a coordinate plane, a vector can be expressed in component form. In two dimensions, we write v = (vₓ, v_y). Here vₓ is the horizontal component and v_y is the vertical component. In three dimensions, we write v = (vₓ, v_y, v_z).

在坐标平面中，向量可以用分量形式表示。在二维中写作 v = (vₓ, v_y)，其中 vₓ 是水平分量，v_y 是垂直分量。在三维中写作 v = (vₓ, v_y, v_z)。

The magnitude of vector v is found using Pythagoras’ Theorem. For v = (vₓ, v_y):

向量 v 的模长可以通过勾股定理求得。对于 v = (vₓ, v_y)：

|v| = √(vₓ² + v_y²)

In three dimensions, add the z-component squared:

在三维情况下，再加上 z 分量的平方：

|v| = √(vₓ² + v_y² + v_z²)

For example, if v = (3, 4), then |v| = √(3² + 4²) = √25 = 5. This is a common starter question in IB Paper 1.

例如，若 v = (3, 4)，则 |v| = √(3² + 4²) = √25 = 5。这是IB卷一试卷中常见的入门题型。

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

A position vector describes the location of a point relative to the origin O. If point A has coordinates (x₁, y₁), then the position vector of A is written as a = (x₁, y₁). Position vectors always start at the origin.

位置向量描述点相对于原点 O 的位置。若点 A 的坐标为 (x₁, y₁)，则 A 的位置向量写作 a = (x₁, y₁)。位置向量总是从原点出发。

The displacement vector from point A to point B is given by subtracting the position vector of A from the position vector of B:

从点 A 到点 B 的位移向量等于点 B 的位置向量减去点 A 的位置向量：

AB = b − a

For example, if A = (1, 2) and B = (4, 6), then AB = (4 − 1, 6 − 2) = (3, 4). The magnitude |AB| gives the distance between A and B.

例如，若 A = (1, 2)，B = (4, 6)，则 AB = (4 − 1, 6 − 2) = (3, 4)。模长 |AB| 就是 A 与 B 之间的距离。

Be careful with order: vector AB points from A to B, while vector BA points from B to A and is exactly the opposite direction. AB = −BA.

务必注意顺序：向量 AB 从 A 指向 B，而向量 BA 从 B 指向 A，方向正好相反。即 AB = −BA。

4. Vector Addition and Scalar Multiplication | 向量加法与标量乘法

Vector addition is performed component by component. If a = (a₁, a₂) and b = (b₁, b₂), then:

向量加法按分量逐一进行。若 a = (a₁, a₂)，b = (b₁, b₂)，则：

a + b = (a₁ + b₁, a₂ + b₂)

Geometrically, place the tail of b at the tip of a; the resultant vector goes from the tail of a to the tip of b. This is the triangle rule of vector addition.

几何上，将 b 的起点放在 a 的终点，连接 a 的起点与 b 的终点，所得向量即为 a + b。这就是向量加法的三角形法则。

Scalar multiplication multiplies each component by the scalar. If k is a scalar, then:

标量乘法将每个分量同时乘以该标量。若 k 为一个实数，则：

k · v = (k·vₓ, k·v_y)

If k > 0, the direction of the vector remains the same; if k < 0, the direction reverses. The magnitude becomes |k| × |v|.

若 k > 0，向量的方向不变；若 k < 0，方向相反。模长变为 |k| × |v|。

5. Negative Vectors and Subtraction | 负向量与向量减法

The negative of a vector v, written −v, has the same magnitude as v but the opposite direction. If v = (vₓ, v_y), then −v = (−vₓ, −v_y).

向量 v 的负向量记作 −v，其模长与 v 相同，但方向相反。若 v = (vₓ, v_y)，则 −v = (−vₓ, −v_y)。

Vector subtraction is defined as adding the negative: a − b = a + (−b). Componentwise:

向量减法定义为加上负向量：a − b = a + (−b)。按分量计算：

a − b = (a₁ − b₁, a₂ − b₂)

Geometrically, if you picture a and b sharing the same tail, then a − b is the vector from the tip of b to the tip of a. This is very useful for finding displacement vectors in geometry questions.

几何上，若 a 与 b 共起点，则 a − b 表示从 b 的终点指向 a 的终点的向量。这在几何题中求位移向量时非常有用。

6. Unit Vectors and Direction | 单位向量与方向

A unit vector is a vector with magnitude exactly 1. It is used to describe direction without size. The unit vector in the direction of v is denoted by v̂ and is calculated by dividing v by its magnitude:

单位向量是模长恰好为 1 的向量，用来描述方向而不涉及大小。与 v 同方向的单位向量记作 v̂，通过将 v 除以它的模长得到：

v̂ = v ÷ |v|

For example, if v = (3, 4), then |v| = 5, so v̂ = (3/5, 4/5). You may also see this written as v / |v|.

例如，若 v = (3, 4)，则 |v| = 5，因此 v̂ = (3/5, 4/5)，也可以写作 v / |v|。

The standard unit vectors along the coordinate axes are often called i, j and k. In two dimensions:

沿坐标轴方向的单位向量通常记作 i、j 和 k。在二维中：

i = (1, 0), j = (0, 1)

Any vector v = (vₓ, v_y) can be written as v = vₓi + v_yj. In IB mark schemes, this notation is accepted as long as you are consistent.

任意向量 v = (vₓ, v_y) 都可以写成 v = vₓi + v_yj。在IB评分标准中，只要表达一致，这种写法是允许的。

7. Parallel, Collinear and Zero Vectors | 平行、共线与零向量

Two vectors a and b are parallel if one is a scalar multiple of the other. That is, a = k·b for some nonzero scalar k. Parallel vectors have the same or opposite direction.

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：a = k·b（k 为非零实数）。平行向量的方向相同或相反。

To test whether two given vectors are parallel, check whether their components are proportional. For example, (2, 4) and (1, 2) are parallel because (2, 4) = 2 × (1, 2).

判断两个向量是否平行，可以检查它们的分量是否成比例。例如，(2, 4) 与 (1, 2) 平行，因为 (2, 4) = 2 × (1, 2)。

Points A, B and C are collinear if the vectors AB and BC are parallel. In practical problems, you can show that AB = k·BC for some scalar k.

点 A、B、C 共线，当且仅当向量 AB 与向量 BC 平行。在解题中，只需证明 AB = k·BC（k 为常数）即可。

The zero vector 0 has magnitude 0 and undefined direction. The sum of a vector and its negative is always the zero vector: v + (−v) = 0.

零向量 0 的模长为 0，方向未定义。任何一个向量与其负向量相加都等于零向量：v + (−v) = 0。

8. Dot Product and Work Done | 点积（数量积）

The dot product, also called the scalar product, is a fundamental operation between two vectors. The dot product of a and b is defined as:

点积，也称为数量积，是两个向量之间最重要的运算之一。向量 a 和 b 的点积定义为：

a · b = |a| |b| cos θ

Here θ is the angle between the two vectors when they are placed tail to tail, with 0° ≤ θ ≤ 180°. The dot product is a scalar, not a vector.

其中 θ 是两个向量共起点时所夹的角，范围是 0° ≤ θ ≤ 180°。点积的结果是一个标量，而不是向量。

If a = (a₁, a₂) and b = (b₁, b₂), the dot product can also be calculated directly from components:

若 a = (a₁, a₂)，b = (b₁, b₂)，则点积也可以直接由分量计算：

a · b = a₁b₁ + a₂b₂

In three dimensions, a · b = a₁b₁ + a₂b₂ + a₃b₃. This formula is quick and avoids needing the angle first.

在三维中，a · b = a₁b₁ + a₂b₂ + a₃b₃。这个公式计算快捷，无需先求出夹角。

A useful physical example is work done: if force F is applied along displacement d, then work W = F · d. This connects pure maths with real-world applications in IB Physics.

一个常见的物理应用是做功：若力 F 沿位移 d 做功，则 W = F · d。这可将纯数学与IB物理中的实际情景联系起来。

9. Finding the Angle Between Two Vectors | 求两向量之间的夹角

Rearranging the dot product formula gives a direct method for finding the angle between two vectors:

将点积公式变形，即可直接求出两个向量之间的夹角：

cos θ = (a · b) ÷ (|a| |b|)

So θ = cos⁻¹[(a · b) ÷ (|a| |b|)]. In IB exams, you must be comfortable using this formula both ways: given components find the angle, or given the angle find the dot product.

因此 θ = cos⁻¹[(a · b) ÷ (|a| |b|)]。在IB考试中，你需要熟练地双向使用这个公式：给出分量求夹角，或给出夹角求点积。

For example, if a = (1, 0) and b = (0, 1), then a · b = 0, |a| = 1, |b| = 1, so cos θ = 0. Therefore θ = 90°. This matches the fact that i and j are perpendicular.

例如，若 a = (1, 0)，b = (0, 1)，则 a · b = 0，|a| = 1，|b| = 1，所以 cos θ = 0，因此 θ = 90°。这与 i 和 j 互相垂直完全吻合。

Remember that the angle between vectors is always taken as the smaller angle between 0° and 180°, so the result of cos⁻¹ is always in this range when found correctly.

注意，两个向量之间的夹角总是取 0° 到 180° 之间的较小角，因此用反余弦求出结果时，角度始终落在这个范围内。

10. Perpendicular Vectors | 垂直向量

Two nonzero vectors a and b are perpendicular if and only if their dot product is zero:

两个非零向量 a 与 b 互相垂直，当且仅当它们的点积为零：

a · b = 0

This is because cos 90° = 0, so the formula a · b = |a| |b| cos θ becomes zero. This condition is frequently used in IB questions to find unknown components.

这是因为 cos 90° = 0，所以点积公式 a · b = |a| |b| cos θ 的结果为零。这一条件在IB题目中常用于求解未知分量。

For example, if a = (2, 3) and b = (x, 4), and a ⊥ b, then 2x + 12 = 0, so x = −6. Checking: a · b = 2(−6) + 3(4) = −12 + 12 = 0.

例如，若 a = (2, 3)，b = (x, 4)，且 a ⊥ b，则 2x + 12 = 0，解得 x = −6。验证：a · b = 2(−6) + 3(4) = −12 + 12 = 0。

Do not confuse perpendicular vectors with parallel vectors. Parallel vectors satisfy a = k·b; perpendicular vectors satisfy a · b = 0. They are completely different conditions.

不要把垂直向量和平行向量混淆。平行向量满足 a = k·b；垂直向量满足 a · b = 0。这是两个完全不同的条件。

11. Using Vectors to Solve Geometry Problems | 用向量解决几何问题

Vectors are powerful tools for proving geometric properties. To prove that AB is parallel to CD, show that AB = k·CD. To prove that two lines are perpendicular, show that their direction vectors have dot product zero.

向量是证明几何性质的有力工具。要证明 AB 平行于 CD，只需证明 AB = k·CD。要证明两条直线垂直，只需证明它们的方向向量点积为零。

To prove that three points A, B and C are collinear, show that AB and BC are parallel and share the common point B. This means AB = k·BC for some scalar k.

要证明 A、B、C 三点共线，需要证明 AB 与 BC 平行，并且它们共有点 B。即存在常数 k，使得 AB = k·BC。

The midpoint M of two points A and B has position vector:

两点 A 和 B 的中点 M 的位置向量为：

m = (a + b) ÷ 2

This is simply the average of the position vectors. It is a fast and reliable method for midpoint problems.

这其实就是位置向量的平均值。对于求中点问题，这是一种快速且可靠的方法。

Vectors can also be used to find the distance between two points, the ratio in which a point divides a line segment, and the area of triangles when combined with the cross product in 3D.

向量还可用于求两点间距离、求点分线段的比例，以及在三维中结合叉积求三角形面积等。

12. Common Pitfalls and Exam Tips | 常见错误与备考建议

Always write the arrow or bold notation to distinguish vectors from scalars. Many IB students lose marks simply because they write quantity without vector notation.

始终使用箭头或粗体符号来区分向量与标量。很多IB学生仅仅因为没有使用向量符号而失分。
Do not confuse AB and BA. AB = −BA, so direction matters. When a question asks for vector from A to B, subtract A from B.

不要混淆 AB 和 BA。AB = −BA，方向很重要。若题目求从 A 到 B 的向量，要用 B 减去 A。
When finding the angle between two vectors, always calculate the dot product first, then divide by the product of the magnitudes. Do not skip the magnitude step.

求两向量夹角时，一定要先算点积，再除以两个模长的乘积。不要跳过求模长的步骤。
Check whether the vector is in two or three dimensions. Missing the z-component is a common arithmetic error in 3D vector problems.

确认向量是二维还是三维。在三维向量题中漏写 z 分量是一个常见的计算错误。
Remember that |v| is always non-negative. If you get a negative value for magnitude, you have made a mistake.

记住 |v| 永远是非负数。如果你算出模长为负，说明一定算错了。
For perpendicular conditions, use a · b = 0, not the cross product. The cross product is for 3D and gives a vector, which is not required in this basic topic.

垂直条件用 a · b = 0，而不是叉积。叉积是三维内容，得到的是向量，不在基础知识点中。
Always draw a quick sketch when possible. A diagram helps you see the direction of the vectors and reduces sign errors.

尽可能快速画一个示意图。图形有助于看清向量方向，减少正负号错误。
Practice converting between component form and magnitude-direction form. In direction form, v = |v|(cos θ, sin θ), where θ is measured from the positive x-axis.

练习在分量形式与“模长+方向”形式之间转换。在方向形式中，v = |v|(cos θ, sin θ)，其中 θ 是从 x 轴正方向开始测量的角度。

By mastering these vector basics step by step, you will build a solid foundation for more advanced topics such as vector equations of lines, dot product applications, and even calculus with vector functions. Consistency in notation and careful arithmetic are the two keys to success in IB vector questions.

只要一步一步掌握这些向量基础，你就能为更深入的内容奠定扎实的基础，例如直线方程的向量形式、点积的应用，甚至含向量函数的微积分。符号表达的一致性和仔细的运算，是在IB向量题中取得高分的关键。

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IB Math: Essential Basics of Vectors | 向量入门必备基础知识梳理