Vectors | 向量

📚 Vectors | 向量

Vectors appear frequently in IGCSE Mathematics, especially in the Extended course. They are used to describe movement, position, and geometric relationships. This guide will help you master vector notation, operations, and problem-solving skills.

向量在 IGCSE 数学中经常出现，尤其是在拓展课程中。它们用于描述运动、位置和几何关系。本指南将帮助你掌握向量符号、运算以及解题技巧。

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1. Vectors and scalars | 向量与标量

A vector is a quantity that has both magnitude and direction. Magnitude means size or length. Direction tells you which way the vector points.

向量是既有大小又有方向的量。大小指的是长度或规模。方向告诉你向量指向哪一边。

A scalar is a quantity that has magnitude only. It does not have a direction. For example, speed is scalar, but velocity is a vector.

标量是只有大小的量。它没有方向。例如，速率是标量，而速度是向量。

Vector quantities	Scalar quantities
Displacement	Distance
Velocity	Speed
Force	Mass
Acceleration	Time

In diagrams, vectors are often drawn as arrows. The length of the arrow shows the magnitude, and the arrowhead shows the direction.

在图中，向量通常用箭头表示。箭头的长度表示大小，箭头指向表示方向。

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2. Vector notation | 向量符号

Vectors can be written in bold type, such as a, or with an arrow above, such as a→. In handwriting, an underline or arrow is common.

向量可以用粗体书写，例如 a，或者在上面加箭头，例如 a→。手写时通常使用下划线或箭头。

A vector between two points A and B can be written as AB→. The first letter is the starting point and the second letter is the end point.

两点 A 和 B 之间的向量可以写成 AB→。第一个字母是起点，第二个字母是终点。

For example, AB→ means the displacement from A to B. The reverse vector is BA→, and it points in the opposite direction.

例如，AB→ 表示从 A 到 B 的位移。反向向量是 BA→，它指向相反方向。

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3. Column vectors | 列向量

A column vector is written with two components: the top number is the horizontal movement, and the bottom number is the vertical movement.

列向量由两个分量书写：上面的数字表示水平移动，下面的数字表示垂直移动。

v = (3, −2)

In this vector, the top component 3 means move 3 units to the right. The bottom component −2 means move 2 units down.

在这个向量中，上面的分量 3 表示向右移动 3 个单位。下面的分量 −2 表示向下移动 2 个单位。

If the top component is negative, move left. If the bottom component is positive, move up.

如果上面的分量是负数，则向左移动。如果下面的分量是正数，则向上移动。

u = (−4, 5)

Here u means move 4 units left and 5 units up. Column vectors are very useful for adding and subtracting.

这里 u 表示向左移动 4 个单位并向上移动 5 个单位。列向量在加法和减法中非常有用。

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4. Equal vectors and negative vectors | 相等向量与相反向量

Two vectors are equal if they have exactly the same magnitude and the same direction. Their starting point does not matter.

两个向量如果大小和方向完全相同，则它们相等。它们的起点并不重要。

Negative vectors have the same magnitude but opposite direction. To find a negative vector, change the sign of each component.

相反向量具有相同的大小但方向相反。要求相反向量，只需改变每个分量的符号。

If v = (3, −1), then −v = (−3, 1)

The vector BA→ is the negative of AB→. So AB→ = −BA→.

向量 BA→ 是 AB→ 的相反向量。因此 AB→ = −BA→。

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5. Addition and subtraction of vectors | 向量的加法与减法

To add column vectors, add the top components together and add the bottom components together.

要将列向量相加，只需分别相加上面的分量和下面的分量。

a = (2, 5), b = (−1, 3) → a + b = (2 + (−1), 5 + 3) = (1, 8)

Subtraction works in the same way. Subtract the components of the second vector from the first vector.

减法的方法相同。用第一个向量的分量减去第二个向量的分量。

a − b = (2 − (−1), 5 − 3) = (3, 2)

Vector addition can also be seen as a triangle. Place the tail of the second vector at the head of the first vector. The sum is the vector from the first tail to the second head.

向量加法也可以看成三角形法则。将第二个向量的尾部放在第一个向量的头部。和向量就是从第一个尾部到第二个头部的向量。

The parallelogram rule gives the same result. Both rules are useful for drawing vector sums.

平行四边形法则给出相同的结果。两种法则在画向量和时都很有用。

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6. Scalar multiplication | 标量乘法

Multiplying a vector by a positive scalar changes its length but keeps its direction the same. Multiply each component by the scalar.

向量乘以正标量会改变其长度但方向不变。将每个分量乘以该标量。

2v = 2 × (3, −2) = (6, −4)

If the scalar is negative, the direction reverses. For example, −2v has the opposite direction to v and twice its length.

如果标量是负数，方向会反转。例如，−2v 的方向与 v 相反，长度是 v 的两倍。

Two vectors are parallel if one is a scalar multiple of the other. This is a very common exam question.

如果一个向量是另一个向量的标量倍数，则这两个向量平行。这是非常常见的考题。

If a = (4, −2) and b = (2, −1), then a = 2b, so a and b are parallel.

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7. Magnitude of a vector | 向量的模

The magnitude of a vector is its length. It is always a non-negative number. Use Pythagoras' theorem to find it.

向量的模是它的长度。它总是非负数。使用勾股定理来求模。

For v = (x, y), |v| = √x² + y²

Example: find the magnitude of v = (3, 4).

示例：求 v = (3, 4) 的模。

|v| = √3² + 4² = √9 + 16 = √25 = 5

sented as AB→, its magnitude is the distance between A and B. Magnitude is also written as |AB→|.

如果向量以 AB→ 的形式给出，它的模就是 A 与 B 之间的距离。模也可以写成 |AB→|。

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8. Position vectors | 位置向量

s at a point P. If point P has coordinates (x, y), then the position vector OP→ is (x, y).

位置向量从原点 O 开始，终点为点 P。如果点 P 的坐标是 (x, y)，则位置向量 OP→ 为 (x, y)。

Position vectors allow us to describe points using vectors. This is very useful in vector geometry.

位置向量让我们可以用向量来表示点。这在向量几何中非常有用。

as position vector OA→ and B has position vector OB→, then the displacement vector AB→ is:

如果 A 的位置向量为 OA→，B 的位置向量为 OB→，则位移向量 AB→ 为：

AB→ = OB→ − OA→ = b − a

is (2, 1) and B is (5, 6). Then a = (2, 1), b = (5, 6), so AB→ = (5 − 2, 6 − 1) = (3, 5).

示例：A 为 (2, 1)，B 为 (5, 6)。则 a = (2, 1)，b = (5, 6)，所以 AB→ = (5 − 2, 6 − 1) = (3, 5)。

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9. Vector geometry | 向量几何

Vectors can prove geometric facts such as midpoints, collinear points, and parallel lines. You need to write clear steps using vector notation.

向量可以证明几何事实，例如中点、共线点和平行线。你需要使用向量符号写出清晰的步骤。

For the midpoint M of AB, its position vector is:

对于 AB 的中点 M，其位置向量为：

OM→ = (OA→ + OB→) / 2

This is the average of the two position vectors. It is often written as m = (a + b) / 2.

这是两个位置向量的平均值。通常写作 m = (a + b) / 2。

a scalar multiple of QR→. For example, if PQ→ = 3QR→, they lie on the same straight line.

如果 PQ→ 是 QR→ 的标量倍数，则点 P、Q、R 共线。例如，如果 PQ→ = 3QR→，它们就在同一条直线上。

Parallel vectors also use scalar multiples. If AB→ = k CD→, then AB is parallel to CD.

平行向量也使用标量倍数关系。如果 AB→ = k CD→，则 AB 平行于 CD。

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10. Common mistakes | 常见错误

• Interchanging the top and bottom components. 混淆上下分量。
• Forgetting that direction matters for vector equality. 忘记向量相等时方向很重要。
• Confusing scalar multiplication with vector addition. 将标量乘法与向量加法混淆。
• Writing a negative magnitude, for example |v| = −5. 把模写成负数，例如 |v| = −5。
• Using incorrect notation such as writing a vector without its arrow or bold type. 使用错误符号，例如写向量时不加箭头或粗体。

Always check the sign of each component, especially when subtracting or multiplying by a negative scalar.

始终检查每个分量的符号，尤其是在减法或乘以负标量时。

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11. Exam tips | 考试技巧

Draw a clear diagram for vector geometry questions. It helps you see the direction and position of each vector.

对于向量几何题，画出清晰的图。它有助于你看清每个向量的方向和位置。

Show all component working. Even if your final answer is wrong, you can still earn method marks.

展示所有分量计算过程。即使最终答案错误，你仍然可以获得方法分。

Check that a magnitude is a positive number or zero. If you get a negative magnitude, check your Pythagoras calculation.

检查模是否为正数或零。如果得到负的模，请检查勾股定理的计算。

llinear vectors, always state the scalar multiple clearly. For example, write PQ→ = 2 RS→.

在证明平行或共线向量时，始终清楚地写出标量倍数。例如，写出 PQ→ = 2 RS→。

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12. Summary | 小结

Vectors have magnitude and direction. Column vectors show horizontal and vertical components. You can add, subtract, and multiply vectors by scalars. The magnitude uses Pythagoras' theorem. Position vectors help solve geometry problems.

向量有大小和方向。列向量显示水平和垂直分量。你可以对向量进行加法、减法和标量乘法。模使用勾股定理计算。位置向量有助于解决几何问题。

Practice with column vectors, magnitude questions, and geometric proofs. Vectors are very structured, so clear working will always help you gain marks.

多做列向量、模和几何证明的练习。向量非常有规律，清晰的解题过程总能帮助你得分。

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