📚 H - Operations with vectors in space | 空间向量的运算
Vectors in three-dimensional space serve as a cornerstone of IB Mathematics, extending familiar two-dimensional concepts into a setting that models physical quantities such as force, velocity, and displacement. Mastering operations with spatial vectors—addition, scalar multiplication, dot and cross products—enables students to solve geometric problems involving angles, areas, volumes, and the relative orientation of lines and planes with precision and elegance.
三维空间中的向量是 IB 数学的基石之一，它把大家熟悉的二维概念扩展到了能够模拟力、速度、位移等物理量的框架中。掌握空间向量的运算——加法、标量乘法、点积和叉积——能让你精确而优雅地解决涉及角度、面积、体积以及线面相对方位的几何问题。
1. Vectors in 3D Space | 三维空间中的向量
A vector in space is a quantity that possesses both magnitude and direction, and it can be represented geometrically as a directed line segment. Unlike scalars, which are fully described by a single number, a vector in three dimensions is identified by three components corresponding to movements along the x‑, y‑ and z‑axes.
空间中的向量是既有大小又有方向的量，在几何上可以用一条有向线段来表示。与只需一个数值就能完全描述的标量不同，三维向量由三个分量确定，分别对应沿 x 轴、y 轴和 z 轴的变化。
When we write a vector as v = (v₁, v₂, v₃), the real numbers v₁, v₂ and v₃ are called the scalar components. The vector can also be expressed using the standard basis vectors i, j, k along the coordinate axes: v = v₁i + v₂j + v₃k.
当我们把向量写成 v = (v₁, v₂, v₃) 时，实数 v₁、v₂ 和 v₃ 称为该向量的标量分量。向量也可以用沿坐标轴的标准基向量 i、j、k 来表示：v = v₁i + v₂j + v₃k。
2. Position Vectors and Free Vectors | 位置向量与自由向量
The position vector of a point P(x, y, z) is the vector OP = xi + yj + zk, anchored at the origin O. A free vector, on the other hand, can be translated anywhere in space without changing its components; two free vectors are equal if their corresponding components are identical.
点 P(x, y, z) 的位置向量是起点固定在原点 O 的向量 OP = xi + yj + zk。而自由向量则可在空间中任意平移且不改变其分量；只要对应分量完全相同，两个自由向量就相等。
The vector that joins two points A(x₁, y₁, z₁) and B(x₂, y₂, z₂) is obtained by subtracting their position vectors: AB = OB − OA = (x₂−x₁, y₂−y₁, z₂−z₁). This provides a direct link between geometry and algebra.
联结两点 A(x₁, y₁, z₁) 和 B(x₂, y₂, z₂) 的向量可由它们的位置向量相减得到：AB = OB − OA = (x₂−x₁, y₂−y₁, z₂−z₁)，这就在几何与代数之间建立了直接的联系。
3. Vector Addition and Subtraction | 向量的加法与减法
Addition of two vectors u = (u₁, u₂, u₃) and v = (v₁, v₂, v₃) is performed component‑wise: u + v = (u₁+v₁, u₂+v₂, u₃+v₃). Geometrically, this corresponds to placing the tail of v at the head of u and drawing the resultant from the tail of u to the head of v (the triangle law).
两个向量 u = (u₁, u₂, u₃) 与 v = (v₁, v₂, v₃) 的加法按分量逐个进行：u + v = (u₁+v₁, u₂+v₂, u₃+v₃)。在几何上，这相当于把 v 的起点移到 u 的终点，然后从 u 的起点到 v 的终点画出合向量（三角形法则）。
Subtraction is defined as adding the opposite vector: u − v = u + (−v) = (u₁−v₁, u₂−v₂, u₃−v₃). The resulting vector points from the head of v to the head of u when the vectors share the same initial point (parallelogram law).
减法定义为加上反向向量：u − v = u + (−v) = (u₁−v₁, u₂−v₂, u₃−v₃)。当两向量起点相同时，差向量的方向由 v 的终点指向 u 的终点（平行四边形法则）。
4. Scalar Multiplication and Parallel Vectors | 标量乘法与平行向量
Multiplying a vector v = (v₁, v₂, v₃) by a scalar k ∈ ℝ gives kv = (kv₁, kv₂, kv₃). The magnitude changes by the factor |k| n remains the same if k > 0 and reverses if k < 0. If k = 0, the result is the zero vector 0 = (0, 0, 0).
向量 v = (v₁, v₂, v₃) 乘以标量 k ∈ ℝ 得到 kv = (kv₁, kv₂, kv₃)。其模长变为原来的 |k| 倍；当 k > 0 时方向不变，k < 0 时方向相反。若 k = 0，结果为零向量 0 = (0, 0, 0)。
Two non‑zero vectors u and v are parallel precisely when one is a scalar multiple of the other, i.e. there exists k ∈ ℝ such that u = kv. For unit vectors in particular, this provides a quick way to test for parallelism.
两个非零向量 u 和 v 平行，当且仅当一个向量是另一个的标量倍数，即存在实数 k 使得 u = kv。尤其对单位向量而言，这提供了一种快速检验平行关系的方法。
5. Magnitude of a Vector | 向量的模
The magnitude (or length) of v = (v₁, v₂, v₃) is the Euclidean norm |v| = √v₁² + v₂² + v₃². It represents the distance from the initial point to the terminal point and is always non‑negative. A vector with magnitude 1 is called a unit vector.
向量 v = (v₁, v₂, v₃) 的模（或长度）是欧几里得范数 |v| = √v₁² + v₂² + v₃²。它表示从起点到终点的距离，始终为非负值。模为 1 的向量称为单位向量。
The distance between two points A and B is the magnitude of the vector AB )² + (y₂−y₁)² + (z₂−z₁)²). This extends the distance formula from two to three dimensions.
两点 A 和 B 之间的距离就是向量 AB 的模：d = √((x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²)。这就将距离公式从二维推广到了三维。
6. Unit Vectors and Direction Cosines | 单位向量与方向余弦
For any non‑zero vector v, a unit vector in the same direction is given by û = v / |v|. Normalising a vector is central to describing direction independently of length, especially when working with dot products and projections.
对任意非零向量 v，与其同向的单位向量为 û = v / |v|。将向量单位化对于脱离长度来单独描述方向至关重要，在运用点积和投影时更是如此。
The direction cosines of v are cos α = v₁/|v|, cos β = v₂/|v|, cos γ = v₃/|v|, where α, β, γ are the angles the vector makes with the positive x‑, y‑ and z‑axes. They satisfy the identity cos² α + cos² β + cos² γ = 1.
v 的方向余弦为 cos α = v₁/|v|、cos β = v₂/|v|、cos γ = v₃/|v|，其中 α、β、γ 分别是向量与正 x 轴、y 轴和 z 轴的夹角。它们满足恒等式 cos² α + cos² β + cos² γ = 1。
7. Scalar (Dot) Product | 数量积（点积）
The dot product of a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃) is defined as a · b + a₂b₂ + a₃b₃. This operation combines two vectors to produce a scalar and is commutative, a · b = b · a.
a = (a₁, a₂, a₃) 与 b = (b₁, b₂, b₃) 的点积定义为 a · b = a₁b₁ + a₂b₂ + a₃b₃。该运算将两个向量组合成一个标量，并且满足交换律 a · b = b · a。
Geometrically, a · b = |a||b| gle between the vectors. This relationship yields a vital formula for the angle: cos θ = (a · b) / (|a||b|). Two vectors are perpendicular (orthogonal) if and only if a · b = 0.
在几何上，a · b = |a||b| cos θ，其中 θ 为两向量之间的夹角。由此可得一个关键的夹角公式：cos θ = (a · b) / (|a||b|)。两向量垂直（正交）当且仅当 a · b = 0。
- Algebraic properties: (ka) · b = k(a · b); a · (b + c) = a · b + a · c.
- 代数性质： (ka) · b = k(a · b)；a · (b + c) = a · b + a · c。
8. Vector (Cross) Product | 向量积（叉积）
The cross product of a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃) is a vector perpendicular to both a and b, defined by a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁). It is anti‑commutative: a × b = −b × a.
a = (a₁, a₂, a₃) 和 b = (b₁, b₂, b₃) 的叉积是一个同时垂直于 a 与 b 的向量，定义为 a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁)。叉积具有反交换性：a × b = −b × a。
The magnitude of the cross product is |a × b| = |a||b| sin θ, which equals the area of the parallelogram spanned by a and b. The direction is given by the right‑hand rule: curling fingers from a to b, the thumb points along a × b.
叉积的模为 |a × b| = |a||b| sin θ，其值恰好等于由 a 和 b 张成的平行四边形的面积。方向由右手定则确定：四指从 a 朝 b 弯曲，大拇指所指即为 a × b 的方向。
- Useful identities: i × j = k, j × k = i, k × i = j; a × a = 0.
- 常用恒等式： i × j = k, j × k = i, k × i = j；a × a = 0。
When evaluated as a determinant, the cross product can be written compactly, though manual expansion is often easier and avoids sign errors.
虽然叉积可写成行列式形式，但直接展开往往更简单且不易出现符号错误。
9. Geometric Applications: Angle, Area and Volume | 几何应用：角度、面积与体积
With the dot and cross products, three fundamental geometric measurements become straightforward.
借助点积与叉积，三个基本的几何度量变得十分简单。
Angle between vectors: cos θ = (a · b) / (|a||b| es between edges of a solid or between a line and a plane when combined with the normal vector.
向量夹角： cos θ = (a · b) / (|a||b|)，0 ≤ θ ≤ π。结合法向量时，可用于计算立体图形的棱边夹角或直线与平面的夹角。
Area of a parallelogram: A = |a × b|; area of a triangle: ½ |a × b|. These are essential when vertices are given in coordinates.
平行四边形面积： A = |a × b|；三角形面积： ½ |a × b|。当顶点以坐标形式给出时，这些公式尤其重要。
Volume of a parallelepiped: the scalar triple product V = |a · (b × c)| gives the volume of the parallelepiped formed by a, b, c. The volume of a tetrahedron is one‑sixth of the corresponding parallelepiped: V = ⅙ |a · (b × c)|.
平行六面体体积： 标量三重积 V = |a · (b × c)| 给出由 a、b、c 构成的平行六面体的体积。四面体的体积为相应平行六面体的六分之一：V = ⅙ |a · (b × c)|。
10. Vector Equations of Lines in Space | 空间直线的向量方程
A line passing through point A with position vector a and parallel to direction vector d has the vector equation r = a + td, t ∈ ℝ. The parameter t moves the point along the line; different t values generate all points on the line.
通过点 A（位置向量 a）且平行于方向向量 d 的直线，其向量方程为 r = a + td，t ∈ ℝ。参数 t 使点沿直线移动，取不同 t 值便可生成直线上所有的点。
ived, provided none of the direction ratios is zero. Operations with vectors allow us to find intersections, distances, and relative orientations of lines and planes elegantly.
化为笛卡儿形式即 (x, y, z) = (x₀, y₀, z₀) + t(d₁, d₂, d₃)，进而可推导出对称式方程 (x−x₀)/d₁ = (y−y₀)/d₂ = (z−z₀)/d₃（假设各方向比均不为零）。利用向量运算，便能优雅地求出交点、距离以及线面之间的相对方位。
11. Solving Problems with Vectors in Space | 利用空间向量解题
When faced with a geometric problem in 3D, a reliable strategy is to identify known points, express relevant vectors in component form, and apply the appropriate product. For angle and projection questions, use the dot product; for area or perpendicularity tests, employ the cross product; for volumes, turn to the scalar triple product.
遇到三维几何问题时，一个可靠的策略是：找出已知点，将相关向量用分量形式表示，然后根据目标选用合适的积。有关角度和投影的问题使用点积；涉及面积或垂直性的验证则用叉积；求体积时则借助标量三重积。
Worked example: Given A(1, 2, 3), B(4, 0, 5) and C(2, 5, 1), find the area of triangle ABC. First form AB = (3, −2, 2) and AC = (1, 3, −2). Then compute AB × AC = ((−2)(−2) − 2·3, 2·1 − 3(−2), 3·3 − (−2)·1) = (4−6, 2+6, 9+2) = (−2, 8, 11). Magnitude = √4+64+121 = √189 = 3√21. Area = ½ × 3√21 = (3√21)/2 square units.
例题：已知 A(1, 2, 3)、B(4, 0, 5) 和 C(2, 5, 1)，求三角形 ABC 的面积。首先写出 AB = (3, −2, 2) 和 AC = (1, 3, −2)。然后计算 AB × AC = ((−2)(−2) − 2·3, 2·1 − 3(−2), 3·3 − (−2)·1) = (4−6, 2+6, 9+2) = (−2, 8, 11)。模为 √4+64+121 = √189 = 3√21。面积 = ½ × 3√21 = (3√21)/2 平方单位。
Systematic practice of such processes builds fluency and confidence for both calculator‑free and calculator‑permitted exam questions.
通过系统练习这些步骤，可以提升流畅度与信心，从容应对允许或不许使用计算器的考试问题。
12. Key Formulas and Summary | 关键公式与总结
| Operation | Formula | Geometric meaning |
| Addition | u + v = (u₁+v₁, u₂+v₂, u₃+v₃) | Resultant vector (triangle/parallelogram law) |
| Scalar multiplication | kv = (kv₁, kv₂, kv₃) | Scaling length, possibly reversing direction |
| Magnitude | |v| = √v₁² + v₂² + v₃² | Length of the vector |
| Unit vector | û = v / |v| | Direction without magnitude |
| Dot product | a · b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b| cos θ | Angle, projection, orthogonality test |
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