📚 Geometric Operations with Vectors | 向量的几何运算
Vectors are fundamental in geometry, allowing us to describe positions, displacements, forces, and many other physical quantities. Geometric operations with vectors include addition, subtraction, scalar multiplication, dot product, cross product (in 3D), and applications such as finding angles, projections, areas, and volumes. This article covers key concepts required for IB Mathematics, focusing on both algebraic and geometric interpretations.
向量是几何学中的基础工具，能够描述位置、位移、力以及许多其他物理量。向量的几何运算包括加法、减法、标量乘法、点积、叉积（三维中）以及求角度、投影、面积和体积等应用。本文涵盖IB数学所需的关键概念，既注重代数运算，也强调几何意义。
1. Vector Addition and Subtraction (Triangle and Parallelogram Rules) | 向量的加法与减法（三角形法则和平行四边形法则）
Vector addition can be performed geometrically using the triangle rule or the parallelogram rule. In the triangle rule, the tail of b is placed at the head of a; the sum a + b is the vector from the tail of a to the head of b. The parallelogram rule draws both vectors from a common origin and constructs a parallelogram – the diagonal from the origin represents a + b.
向量加法可以通过三角形法则或平行四边形法则几何作图。在三角形法则中，将向量 b 的尾部放在 a 的头部，和向量 a + b 从 a 的尾部指向 b 的头部。平行四边形法则则从同一起点画两个向量，构成平行四边形，其对角线即为 a + b。
Vector subtraction is equivalent to adding the negative: a – b = a + (–b). Graphically, if a and b are placed tail-to-tail, then a – b is the vector from the head of b to the head of a.
向量减法相当于加上相反向量：a – b = a + (–b)。几何上，若 a 与 b 尾尾相接，则 a – b 是从 b 的头部指向 a 的头部的向量。
In component form, if a = (a₁, a₂) and b = (b₁, b₂) in two dimensions, then:
a + b = (a₁ + b₁, a₂ + b₂), a – b = (a₁ – b₁, a₂ – b₂)
在分量形式下，若二维中 a = (a₁, a₂) 且 b = (b₁, b₂)，则 a + b = (a₁ + b₁, a₂ + b₂)，a – b = (a₁ – b₁, a₂ – b₂)。三维情况可类似地添加第三分量。
Addition is commutative and associative: a + b = b + a; (a + b) + c = a + (b + c). The zero vector 0 satisfies a + 0 = a, and a + (–a) = 0.
加法满足交换律与结合律：a + b = b + a；( a + b ) + c = a + ( b + c )。零向量 0 满足 a + 0 = a，同时 a + (–a) = 0。
2. Scalar Multiplication and Vector Scaling | 标量乘法与向量缩放
Multiplying a vector v by a scalar k produces a vector kv whose magnitude is |k| times |v| ins the same; if k < 0, the direction is reversed. The zero vector is obtained when k = 0.
向量 v 乘以标量 k 得到 kv，其模长是原模长的 |k| 倍。当 k > 0 时方向保持不变；当 k < 0 时方向相反。k = 0 时得到零向量。
In components: kv peration stretches or shrinks the vector and is fundamental for defining parallel vectors.
在分量形式下：kv = (kv₁, kv₂, kv₃)。这一运算可以拉伸或缩短向量，是定义平行向量的基础。
Two non-zero vectors a and b are parallel if and only if there exists a scalar k such that a = kb. This proportionality of components is a crucial geometric condition.
两个非零向量 a 与 b 平行，当且仅当存在标量 k 使得 a = kb。各分量成比例这一条件是一个重要的几何判据。
3. Magnitude of a Vector | 向量的模
The magnitude (or length) of a vector v = (v₁, v₂) in 2D is given by |v| = √v₁² + v₂². In three dimensions, |v| = √v₁² + v₂² + v₃². This represents the Euclidean distance from the origin to the point defined by the vector's coordinates.
二维向量 v = (v₁, v₂) 的模（长度）由 |v| = √v₁² + v₂² 给出。在三维中，|v| = √v₁² + v₂² + v₃²。它表示从原点到向量坐标点的欧几里得距离。
For a displacement vector AB⃗, where A and B have position vectors a and b, the distance between A and B is |b – a| = √((b₁–a₁)² + (b₂–a₂)² + (b₃–a₃)²).
对于位移向量 AB⃗，若 A、B 的位置向量为 a、b，则 A 与 B 之间的距离为 |b – a| = √((b₁–a₁)² + (b₂–a₂)² + (b₃–a₃)²)。
Properties: |v| ≥ 0, with equality only for the zero vector; |kv| = |k| |v|.
性质：|v| ≥ 0，仅当零向量时取等；|kv| = |k| |v|。
4. Unit Vectors and Direction | 单位向量与方向
A unit vector has magnitude 1. The unit vector in the direction of a non-zero vector v is û = v / |v|. It preserves direction while normalizing length.
单位向量的模为1。非零向量 v 方向上的单位向量为 û = v / |v|。它在保留方向的同时将长度单位化。
Standard basis vectors are often used: in 2D, i = (1, 0) and j = (0, 1); in 3D, i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1). Any vector can be expressed as a combination of these, e.g., v = v₁ i + v₂ j + v₃ k.
常用标准基向量：二维中 i = (1, 0), j = (0, 1)；三维中 i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1)。任何向量均可表示为这些基向量的组合，如 v = v₁ i + v₂ j + v₃ k。
Finding a unit vector helps in specifying directions without regard to magnitude, essential for vector projection and angle calculations.
求单位向量有助于无需考虑大小而明确方向，对于向量投影和夹角计算至关重要。
5. Position Vectors and Displacement | 位置向量与位移
The position vector of a point P is the vector from the origin O to P, often denoted by p. If P has coordinates (x, y, z), then p = (x, y, z) or xi + yj + zk.
点 P 的位置向量是从原点 O 到点 P 的向量，通常记为 p。若 P 点坐标为 (x, y, z)，则 p = (x, y, z) 或 xi + yj + zk。
The displacement vector from point A to point B is AB⃗ = b – a, where a and b are the position vectors of A and B. Displacement is independent of the choice of origin.
从点 A 到点 B 的位移向量为 AB⃗ = b – a，其中 a 和 b 是 A、B 的位置向量。位移与原点选择无关。
This concept links coordinate geometry with vector algebra: the midpoint of AB has position vector (a + b)/2, and points on the line AB can be expressed as a + t(b – a) for a scalar t.
这一概念将坐标几何与向量代数联系起来：AB 中点的位置向量为 (a + b)/2，而直线 AB 上的点可表示为 a + t(b – a)，t 为标量。
6. Collinear and Parallel Vectors | 共线向量与平行向量
Points A, B, C are collinear if vectors AB⃗ and BC⃗ are parallel, i.e., AB⃗ = k BC⃗ for some scalar k. Equivalently, the position vectors must satisfy c – a = λ (b – a).
若向量 AB⃗ 与 BC⃗ 平行，即 AB⃗ = k BC⃗ 对某个标量 k 成立，则点 A、B、C 共线。等价地，位置向量必须满足 c – a = λ (b – a)。
Parallel vectors in general may have different positions. Two vectors u and v are parallel if u = mv for a non-zero scalar m. In component terms, corresponding components must be in the same ratio.
一般而言，平行向量可以有不同位置。两个向量 u 与 v 平行，若存在非零标量 m 使得 u = mv。在分量层面，对应分量之比必须相等。
The direction vectors of parallel lines are scalar multiples of each other, which is widely used in line and plane geometry.
平行线的方向向量互为标量倍数，这在线与平面几何中应用广泛。
7. Dot Product and Orthogonality | 点积与正交性
The dot product (scalar product) of two vectors a and b is defined as:
a · b = |a| |b| cos θ
where θ is the angle between a and b (0 ≤ θ ≤ π). In component form, for 3D vectors a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃): a · b = a₁b₁ + a₂b₂ + a₃b₃.
两个向量 a 和 b 的点积（标量积）定义为 a · b = |a| |b| cos θ，其中 θ 是 a 与 b 之间的夹角（0 ≤ θ ≤ π）。在分量形式下，对于三维向量 a = (a₁, a₂, a₃) 和 b = (b₁, b₂, b₃)，a · b = a₁b₁ + a₂b₂ + a₃b₃。
Key properties: commutativity (a · b = b · <
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