Vectors in 3D | 三维向量

📚 Vectors in 3D | 三维向量

Vectors in three dimensions extend every concept you have learnt in 2D into the extra z‑axis. The Edexcel A‑Level Mathematics specification expects you to be confident with vector notation, magnitude, dot product, and the vector equation of a line, all applied in a 3D coordinate system. These ideas appear regularly in Pure Paper 2 and form the backbone of mechanics and further pure topics.

三维向量将你在二维中学到的每一个概念扩展到额外的 z 轴。Edexcel A‑Level 数学大纲要求你熟练掌握向量表示法、模、点积以及直线的向量方程，并能在三维坐标系中加以应用。这些内容经常出现在 Pure Paper 2 中，同时也是力学和进阶纯数的基础。

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1. Vectors in 3D Coordinates | 三维坐标系中的向量表示

A 3D vector is written in component form using the unit vectors i, j and k along the x, y and z axes respectively. Any vector v can be expressed as v = xi + yj + zk or as a column vector (x, y, z)ᵀ. The coordinates of a point (x, y, z) give the position vector relative to the origin.

三维向量用沿 x、y、z 轴的单位向量 i、j、k 以分量形式表示。任意向量 v 可写作 v = xi + yj + zk 或列向量 (x, y, z)ᵀ。点 (x, y, z) 的坐标给出了相对于原点的位置向量。

A vector from point A(x₁, y₁, z₁) to B(x₂, y₂, z₂) is AB = (x₂−x₁)i + (y₂−y₁)j + (z₂−z₁)k. This displacement is independent of the origin.

从点 A(x₁, y₁, z₁) 到 B(x₂, y₂, z₂) 的向量为 AB = (x₂−x₁)i + (y₂−y₁)j + (z₂−z₁)k。该位移与原点无关。

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2. Magnitude of a Vector | 向量的模

The magnitude (length) of a vector v = ai + bj + ck is given by the three‑dimensional extension of Pythagoras’ theorem: |v| = √(a² + b² + c²). It is always non‑negative and represents the distance from the origin if v is a position vector.

向量 v = ai + bj + ck 的模（长度）由毕达哥拉斯定理的三维推广给出：|v| = √(a² + b² + c²)。模总是非负的；如果 v 是位置向量，它就表示到原点的距离。

For example, the vector 3i − 4j + 12k has magnitude √(3² + (−4)² + 12²) = √(9 + 16 + 144) = √169 = 13.

例如，向量 3i − 4j + 12k 的模为 √(3² + (−4)² + 12²) = √(9 + 16 + 144) = √169 = 13。

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3. Unit Vectors and Base Vectors | 单位向量与基向量

A unit vector has magnitude 1. To find the unit vector in the direction of a given vector v, divide the vector by its magnitude: û = v / |v|. Unit vectors are used to indicate direction without scaling.

单位向量的模为 1。要找到给定向量 v 方向上的单位向量，将向量除以其模：û = v / |v|。单位向量用于表示方向而不涉及大小。

The standard base vectors are i = (1, 0, 0), j = (0, 1, 0) and k = (0, 0, 1). Any 3D vector is a unique linear combination of these three.

标准基向量为 i = (1, 0, 0)、j = (0, 1, 0) 与 k = (0, 0, 1)。任何三维向量都是这三个基向量的唯一线性组合。

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4. Position Vectors | 位置向量

The position vector of a point P(x, y, z) is the vector from the origin O to P, written as OP = xi + yj + zk. All points in 3D can be described by their position vectors.

点 P(x, y, z) 的位置向量是从原点 O 到 P 的向量，记作 OP = xi + yj + zk。三维空间中的所有点都可用其位置向量描述。

If A and B have position vectors a and b, then AB = b − a. This relationship is crucial for working with lines and relative positions.

若 A 和 B 的位置向量分别为 a 和 b，则 AB = b − a。该关系在处理直线和相对位置时至关重要。

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5. Distance and Midpoint | 两点间距离与中点

The distance between two points A(x₁, y₁, z₁) and B(x₂, y₂, z₂) is the magnitude of AB: d = √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²].

两点 A(x₁, y₁, z₁) 与 B(x₂, y₂, z₂) 之间的距离就是向量 AB 的模：d = √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²]。

The midpoint M of AB has coordinates ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2), which is the average of the position vectors: m = (a + b)/2.

AB 的中点 M 的坐标为 ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2)，它等于位置向量的平均值：m = (a + b)/2。

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

Addition of two 3D vectors is done component‑by‑component: (a₁i+b₁j+c₁k) + (a₂i+b₂j+c₂k) = (a₁+a₂)i + (b₁+b₂)j + (c₁+c₂)k. Vector addition is commutative and associative.

三维向量加法按分量进行：(a₁i+b₁j+c₁k) + (a₂i+b₂j+c₂k) = (a₁+a₂)i + (b₁+b₂)j + (c₁+c₂)k。向量加法满足交换律和结合律。

Scalar multiplication multiplies each component by the scalar: λ(ai+bj+ck) = λai + λbj + λck. When λ is negative, the vector reverses direction.

标量乘法将每个分量乘以该标量：λ(ai+bj+ck) = λai + λbj + λck。当 λ 为负数时，向量的方向相反。

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7. Dot Product / Scalar Product | 点积/标量积

The dot product of two vectors u = a₁i+b₁j+c₁k and v = a₂i+b₂j+c₂k is defined as u · v = a₁a₂ + b₁b₂ + c₁c₂. The result is a scalar, not a vector.

两个向量 u = a₁i+b₁j+c₁k 与 v = a₂i+b₂j+c₂k 的点积定义为 u · v = a₁a₂ + b₁b₂ + c₁c₂。结果是一个标量，而非向量。

The geometric interpretation is u · v = |u||v| cos θ, where θ is the angle between the vectors. This formula is central for finding angles and testing perpendicularity.

点积的几何意义为 u · v = |u||v| cos θ，其中 θ 是两向量之间的夹角。这个公式是求角度和判断垂直的核心。

Key properties: u · v = v · u, and u · u = |u|².

主要性质：u · v = v · u，且 u · u = |u|²。

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8. Finding the Angle Using Dot Product | 利用点积求角度

From u · v = |u||v| cos θ, the angle between two non‑zero vectors is given by cos θ = (u · v) / (|u||v|). Always check that the vectors are not zero before computing.

由 u · v = |u||v| cos θ 可得，两非零向量之间的夹角满足 cos θ = (u · v) / (|u||v|)。计算前务必确认向量非零。

For example, using a = 2i + j − 2k and b = i + 3j + 4k: a·b = 2×1 + 1×3 + (−2)×4 = 2+3−8 = −3; |a| = 3, |b| = √26, so cos θ = −3/(3√26) = −1/√26, giving θ ≈ 101.3°.

例如，取 a = 2i + j − 2k 和 b = i + 3j + 4k：a·b = 2×1 + 1×3 + (−2)×4 = −3；|a| = 3，|b| = √26，故 cos θ = −3/(3√26) = −1/√26，得 θ ≈ 101.3°。

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9. Parallel and Perpendicular Vectors | 平行向量与垂直向量

Two non‑zero vectors are parallel if one is a scalar multiple of the other: u = λv. This means their components are proportional. They are perpendicular (orthogonal) if their dot product is zero: u · v = 0.

若一个非零向量是另一个的标量倍，则两者平行：u = λv。这意味着它们的分量成比例。若点积为零，则两向量垂直（正交）：u · v = 0。

These two tests are vital when working with geometric shapes or lines. For instance, to show that triangle ABC is right‑angled at B, verify that BA · BC = 0.

在处理几何图形或直线时，这两个判据至关重要。例如，要证明三角形 ABC 在 B 处为直角，可验证 BA · BC = 0。

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

In 3D, a straight line can be described by r = a + tb (t ∈ ℝ), where a is the position vector of a fixed point on the line and b is a direction vector parallel to the line. Changing t moves you along the line.

在三维空间中，一条直线可表示为 r = a + tb（t ∈ ℝ），其中 a 是直线上一个固定点的位置向量，b 是一个与直线平行的方向向量。改变 t 就在直线上移动。

The parametric form consists of three component equations: x = a₁ + t b₁, y = a₂ + t b₂, z = a₃ + t b₃. The Cartesian form eliminates t: (x−a₁)/b₁ = (y−a₂)/b₂ = (z−a₃)/b₃, provided none of b₁, b₂, b₃ is zero.

参数式由三个分量方程组成：x = a₁ + t b₁，y = a₂ + t b₂，z = a₃ + t b₃。将 t 消去即得笛卡儿式：(x−a₁)/b₁ = (y−a₂)/b₂ = (z−a₃)/b₃，前提是 b₁、b₂、b₃ 均不为零。

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11. Relationships Between Lines: Intersecting, Skew, Parallel | 直线间的关系：相交、歪斜与平行

Two lines in 3D can be parallel, intersecting, or skew (neither parallel nor intersecting). To decide, first compare direction vectors: if b₁ = λb₂ the lines are parallel (or coincident if they share a point).

三维空间中的两条直线可能平行、相交或歪斜（既不平行也不相交）。判断时，先比较方向向量：若 b₁ = λb₂，则两线平行（如果还共享一个点，则重合）。

If the direction vectors are not multiples, set up a system of equations by equating the two vector forms with parameters s and t. Solve two equations for s and t, then check if the solution satisfies the third equation. If it does, lines intersect; if not, they are skew.

若方向向量不成比例，则用两个不同的参数 s 与 t 令两向量式相等，建立方程组。先解其中两个方程求出 s 与 t，再代入第三个方程检验。若满足，则相交；否则为歪斜。

For intersecting lines, the point of intersection can be found by substituting t or s back into the respective line equation.

对相交直线，将求得的 t 或 s 代回相应直线方程，即可求出交点。

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12. Collinear Points and Ratios | 共线点与线段比例

Three points A, B, C are collinear if the vectors AB and AC are parallel, i.e. AB = kAC for some scalar k. This also implies that one point lies on the line through the other two.

若向量 AB 与 AC 平行，即存在标量 k 使得 AB = kAC，则三点 A、B、C 共线。这也意味着其中一个点在另外两点决定的直线上。

You can use position vectors and a ratio to divide a line segment internally or externally. For internal division in the ratio m:n, the position vector of the point is given by p = (na + mb) / (m + n).

利用位置向量和比例，可以内分或外分一条线段。对于按 m:n 内分的点，其位置向量为 p = (na + mb) / (m + n)。

Such ratios appear frequently in geometric vector problems that ask for a point on a line at a given proportion.

这类比例经常出现在要求按给定比例求直线上一点的几何向量问题中。

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