Vectors | 向量

📚 Vectors | 向量

In Edexcel A-Level Mathematics, vectors describe quantities that have both magnitude and direction, such as displacement, velocity, and force. This topic combines coordinate geometry, algebra, and trigonometry, and it appears in both pure and applied contexts.

在爱德思 A-Level 数学中，向量用来描述既有大小又有方向的量，例如位移、速度和力。该主题综合了坐标几何、代数与三角学，并在纯数学和应用数学中都会出现。

1. Vector Notation and Representation | 向量表示法

A vector can be written in component form using the standard basis vectors i, j, and k. In two dimensions, a = xi + yj. In three dimensions, a = xi + yj + zk.

向量可以用标准基向量 i、j、k 的分量形式表示。在二维中，a = xi + yj；在三维中，a = xi + yj + zk。

The same vector may also be represented as a column vector or as a directed line segment from an initial point to a terminal point. Bold letters or underlined letters are commonly used in handwriting.

同一个向量也可以表示为列向量或从起点到终点的有向线段。手写时通常用粗体字母或下划线字母表示向量。

2D form	a = xi + yj
3D form	a = xi + yj + zk
Column form	(x, y) in 2D or (x, y, z) in 3D

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

A position vector is the vector from the origin O to a point A, written as OA = a. If point B has position vector b, then the displacement vector from A to B is AB = OB − OA = b − a.

位置向量是从原点 O 到点 A 的向量，记作 OA = a。如果点 B 的位置向量为 b，那么从 A 到 B 的位移向量为 AB = OB − OA = b − a。

This result is essential for finding vectors between two points in coordinate geometry problems. Always subtract the starting point’s position vector from the ending point’s position vector.

这个结果对于在坐标几何问题中求解两点之间的向量至关重要。一定要用终点位置向量减去起点位置向量。

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3. Magnitude and Direction of a Vector | 向量的大小与方向

For a 2D vector a = xi + yj, the magnitude is |a| = √(x² + y²). In 3D, |a| = √(x² + y² + z²). Magnitude is always non-negative.

对于二维向量 a = xi + yj，其大小为 |a| = √(x² + y²)。在三维中，|a| = √(x² + y² + z²)。向量的大小始终是非负的。

The direction of a 2D vector is described by the angle θ it makes with the positive x-axis. This satisfies tan θ = y ÷ x, but the quadrant of θ must be determined from the signs of x and y.

二维向量的方向由它与正 x 轴所成的角 θ 来描述。该角满足 tan θ = y ÷ x，但 θ 的象限必须根据 x 和 y 的符号来确定。

A unit vector in the direction of a is given by a ÷ |a|. Unit vectors have magnitude 1 and are used to describe direction only.

沿 a 方向的单位向量为 a ÷ |a|。单位向量的大小为 1，仅用于描述方向。

|a| = √(x² + y²) in 2D

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4. Vector Addition and Scalar Multiplication | 向量加法与数乘

Vectors are added component-wise. If a = x₁i + y₁j and b = x₂i + y₂j, then a + b = (x₁ + x₂)i + (y₁ + y₂)j.

向量按分量逐项相加。如果 a = x₁i + y₁j，b = x₂i + y₂j，则 a + b = (x₁ + x₂)i + (y₁ + y₂)j。

Geometrically, addition follows the triangle law or parallelogram law. Scalar multiplication λa changes the magnitude by a factor |λ| and reverses direction if λ is negative.

从几何上看，向量加法遵循三角形法则或平行四边形法则。数乘 λa 将大小改变 |λ| 倍，如果 λ 为负则方向相反。

λa = (λx)i + (λy)j + (λz)k

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5. Parallel Vectors and Collinear Points | 平行向量与共线点

Two vectors a and b are parallel if one is a scalar multiple of the other: a = λb. This also means their component ratios are equal.

两个向量 a 和 b 平行，如果其中一个是另一个的数乘：a = λb。这也意味着它们的对应分量之比相等。

Three points A, B, and C are collinear if the vectors AB and BC are parallel, so AB = λBC for some scalar λ. Collinearity often appears in exam questions asking for an unknown coordinate.

三点 A、B、C 共线，如果向量 AB 和 BC 平行，即 AB = λBC（λ 为某标量）。共线问题常出现在求未知坐标的考题中。

AB = λBC ⇒ A, B, C are collinear

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6. Ratio Theorem and Dividing a Line Segment | 定比分点公式与线段分割

If point P divides the line segment AB in the ratio m : n, then the position vector of P is p = (na + mb) ÷ (m + n).

如果点 P 以 m : n 的比例分割线段 AB，则点 P 的位置向量为 p = (na + mb) ÷ (m + n)。

This formula is derived from AP = (m ÷ (m + n)) AB. It is particularly useful when working with position vectors rather than coordinates.

该公式由 AP = (m ÷ (m + n)) AB 推导而来。在处理位置向量而非坐标时，这个公式尤其有用。

For the midpoint, m = n = 1, so p = (a + b) ÷ 2.

对于中点，m = n = 1，因此 p = (a + b) ÷ 2。

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7. Vector Equation of a Straight Line | 直线的向量方程

A straight line in vector form is written as r = a + td, where a is the position vector of a point on the line, d is a direction vector parallel to the line, and t is a scalar parameter.

直线的向量方程写作 r = a + td，其中 a 是直线上一点的位置向量，d 是与直线平行的方向向量，t 是标量参数。

Different values of t produce different points on the line. The same line can be represented by infinitely many vector equations because any point on the line and any parallel direction vector may be used.

不同的 t 值会产生直线上的不同点。同一条直线可以用无穷多个向量方程表示，因为可以选择直线上任意一点和任意平行的方向向量。

r = a + td, t ∈ ℝ

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8. Cartesian Form of a Line | 直线的笛卡尔方程

If a line passes through (x₁, y₁, z₁) and has direction vector d = d₁i + d₂j + d₃k, its Cartesian form is (x − x₁) ÷ d₁ = (y − y₁) ÷ d₂ = (z − z₁) ÷ d₃.

如果一条直线经过点 (x₁, y₁, z₁) 且方向向量为 d = d₁i + d₂j + d₃k，则其笛卡尔方程为 (x − x₁) ÷ d₁ = (y − y₁) ÷ d₂ = (z − z₁) ÷ d₃。

In two dimensions, only the first two parts are used. If any direction component is zero, that coordinate is constant; for example, if d₁ = 0, then x = x₁.

在二维中，只使用前两部分。如果某个方向分量为零，则对应坐标为常数；例如如果 d₁ = 0，则 x = x₁。

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9. Scalar Product or Dot Product | 标量积或点积

The scalar product of two vectors a = x₁i + y₁j + z₁k and b = x₂i + y₂j + z₂k is a · b = x₁x₂ + y₁y₂ + z₁z₂.

两个向量 a = x₁i + y₁j + z₁k 和 b = x₂i + y₂j + z₂k 的标量积为 a · b = x₁x₂ + y₁y₂ + z₁z₂。

The scalar product is also given by a · b = |a||b| cos θ, where θ is the angle between the two vectors when they are placed tail to tail.

标量积还可表示为 a · b = |a||b| cos θ，其中 θ 是两个向量首尾相接放置时的夹角。

a · b = x₁x₂ + y₁y₂ + z₁z₂ = |a||b| cos θ

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10. Angle Between Vectors and Perpendicularity | 向量夹角与垂直

The angle θ between two non-zero vectors is found using cos θ = (a · b) ÷ (|a||b|). This works in both 2D and 3D.

两个非零向量之间的夹角 θ 可由 cos θ = (a · b) ÷ (|a||b|) 求得。该公式在二维和三维中都适用。

Two vectors are perpendicular if and only if a · b = 0. Parallel vectors give a · b = ±|a||b|, depending on whether they point in the same or opposite directions.

两个向量垂直当且仅当 a · b = 0。平行向量满足 a · b = ±|a||b|，取决于它们方向相同还是相反。

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11. Exam-Style Applications and Common Pitfalls | 考试型应用与常见易错点

Typical Edexcel questions ask for the vector equation of a line through two points, the angle between two lines, or the value of a parameter that makes two vectors perpendicular. Start by finding a direction vector from the two points, then substitute into r = a + td.

爱德思的典型考题包括求经过两点的直线向量方程、求两条直线的夹角，或求使两个向量垂直的参数值。解题时可先由两点求出方向向量，再代入 r = a + td。

A common mistake is confusing the direction vector with a position vector. Another is using the dot product without first checking that the vectors are in the same dimension. Always state the value of λ when proving collinearity rather than just writing the vector equation.

常见错误是将方向向量与位置向量混淆。另一个错误是在未确认向量维数相同的情况下使用点积。证明共线时，一定要写出 λ 的值，而不仅仅是写出向量方程。

For example, given A(1, 2, 3) and B(4, 6, 8), the direction vector is AB = 3i + 4j + 5k, so one line equation is r = i + 2j + 3k + t(3i + 4j + 5k).

例如，已知 A(1, 2, 3) 和 B(4, 6, 8)，方向向量为 AB = 3i + 4j + 5k，因此直线方程之一为 r = i + 2j + 3k + t(3i + 4j + 5k)。

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