📚 Working with Vectors | 向量运算
Vectors are fundamental in mathematics and physics for describing quantities that have both magnitude and direction. In IB Mathematics, you will learn to perform operations with vectors, use them to represent lines and planes, and apply dot and cross products to solve geometric problems.
向量在数学和物理学中用于描述既有大小又有方向的量。在 IB 数学课程中，你将学习向量的运算、用向量表示直线和平面，并运用点积和叉积解决几何问题。
1. Introduction to Vectors | 向量简介
A vector is a quantity defined by both magnitude and direction, such as displacement, velocity or force. This contrasts with a scalar, which has only magnitude, e.g., speed, distance, mass.
向量是由大小和方向共同定义的量，例如位移、速度或力。这与标量不同，标量只有大小，如速率、距离、质量。
Vectors can be represented geometrically as arrows, where the length represents the magnitude and the arrowhead shows the direction. Algebraically, vectors are written in component form, such as v = (x, y) in two dimensions or v = (x, y, z) in three dimensions.
向量可以用几何箭头表示，箭头的长度表示大小，箭头指向表示方向。在代数上，向量以分量形式书写，如在二维中 v = (x, y)，在三维中 v = (x, y, z)。
You will also encounter unit vectors i, j (and k in 3D), which point along the coordinate axes. Any vector can be expressed as a sum of scalar multiples of these base vectors: v = xi + yj (+ zk).
你还会遇到单位向量 i、j（三维中还有 k），它们指向坐标轴的正方向。任何向量都可以表示为这些基向量的标量倍数之和：v = xi + yj（+ zk）。
2. Vector Addition and Subtraction | 向量的加法和减法
Vectors are added geometrically using the triangle law or parallelogram law: placing the tail of one vector at the head of the other. Algebraically, you simply add corresponding components: if a = (a₁, a₂) and b = (b₁, b₂), then a + b = (a₁ + b₁, a₂ + b₂).
向量加法在几何上遵循三角形法则或平行四边形法则：将一个向量的起点放在另一个向量的终点上。在代数上，只需将对应的分量相加：如果 a = (a₁, a₂)，b = (b₁, b₂)，则 a + b = (a₁ + b₁, a₂ + b₂)。
Subtraction is defined as adding the opposite vector: a - b = a + (-b). In components, a - b = (a₁ - b₁, a₂ - b₂). The vector connecting two points A and B is AB = b - a, where a and b are position vectors of A and B.
减法定义为加上反向向量：a - b = a + (-b)。用分量表示为 a - b = (a₁ - b₁, a₂ - b₂)。连接两点 A 和 B 的向量 AB = b - a，其中 a 与 b 分别是 A 和 B 的位置向量。
3. Scalar Multiplication | 标量乘法（数乘）
Multiplying a vector by a scalar k changes its magnitude by |k| and, if k is negative, reverses its direction. In components: kv = (k·x, k·y, k·z). The resulting vector is always parallel to the original.
将向量乘以标量 k 会使其大小变为原来的 |k| 倍；若 k 为负数，方向也会反向。用分量表示：kv = (k·x, k·y, k·z)。其结果向量总与原始向量平行。
Two non‑zero vectors are parallel if one is a scalar multiple of the other: a = λb, λ ∈ ℝ. This condition is extremely useful for proving collinearity and solving geometric problems.
两个非零向量平行，当且仅当其中一个为另一个的标量倍数：a = λb（λ ∈ ℝ）。这一条件在证明共线性和解决几何问题时极为有用。
4. Magnitude of a Vector | 向量的大小（模）
The magnitude (or length) of a vector v = (x, y) is given by |v| = √x² + y². In three dimensions, for v = (x, y, z), |v| = √x² + y² + z². This follows directly from Pythagoras' theorem.
向量 v = (x, y) 的大小（模）为 |v| = √x² + y²。在三维中，若 v = (x, y, z)，则 |v| = √x² + y² + z²。这直接源自勾股定理。
The distance between two points A(x₁, y₁) and B(x₂, y₂) is the magnitude of vector AB: |AB| = √((x₂ - x₁)² + (y₂ - y₁)²). This extends naturally to 3D by including the z‑difference.
两点 A(x₁, y₁) 与 B(x₂, y₂) 之间的距离就是向量 AB 的模：|AB| = √((x₂ - x₁)² + (y₂ - y₁)²)。此公式加入 z 坐标差后同样适用于三维空间。
5. Unit Vectors | 单位向量
A unit vector has magnitude 1. To obtain a unit vector in the direction of any nonzero vector v, divide it by its own magnitude: û = v / |v|. This process is called normalisation.
单位向量的大小为 1。要得到沿任意非零向量 v 方向的单位向量，可将其除以其模：û = v / |v|。这个过程称为单位化。
The standard basis vectors i = (1, 0, 0), j = (0, 1, 0) and k = (0, 0, 1) are mutually perpendicular unit vectors that form a right‑handed system in 3D.
标准基向量 i = (1, 0, 0)、j = (0, 1, 0) 与 k = (0, 0, 1) 是两两垂直的单位向量，在三维中构成右手系。
6. Position Vectors | 位置向量
A position vector describes the location of a point relative to the origin. For a point P(x, y, z), its position vector is OP = p = (x, y, z). It is often denoted simply by p.
位置向量描述了点相对于原点的位置。对于点 P(x, y, z)，其位置向量为 OP = p = (x, y, z)，通常简记为 p。
The vector from point A to point B is obtained by subtracting position vectors: AB = b - a. The midpoint M of AB has position vector (a + b)/2.
从点 A 到点 B 的向量可以通过位置向量相减得到：AB = b - a。线段 AB 的中点 M 的位置向量为 (a + b)/2。
These relationships are independent of the choice of coordinate system and form the backbone of vector geometry proofs.
这些关系与坐标系的选择无关，是向量几何证明的基础。
7. Dot Product (Scalar Product) | 点积（数量积）
The dot product of two vectors a and b is defined as a · b = |a||b| cos θ, where θ is the angle between them. It is a scalar quantity.
两个向量 a 与 b 的点积定义为 a · b = |a||b| cos θ，其中 θ 为二者间的夹角。点积是一个标量。
In component form, if a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃), then a · b = a₁b₁ + a₂b₂ + a₃b₃. The dot product is commutative and distributive over addition.
用分量表示，若 a = (a₁, a₂, a₃) 且 b = (b₁, b₂, b₃)，则 a · b = a₁b₁ + a₂b₂ + a₃b₃。点积满足交换律和对加法的分配律。
A most important property: two nonzero vectors are perpendicular if and only if a · b = 0. This is frequently used to test orthogonality.
最重要的性质：两个非零向量垂直，当且仅当 a · b = 0。这一性质常用来检验垂直关系。
8. Angle Between Two Vectors | 两向量间的夹角
Rearranging the dot product formula gives cos θ = (a · b) / (|a||b|). This allows you to find the angle between any two nonzero vectors.
将点积公式变形可得 cos θ = (a · b) / (|a||b|)。这用于计算任意两个非零向量之间的夹角。
are orthogonal. If θ = 0° or 180°, the vectors are parallel; the dot product then equals ±|a||b|.
当 θ = 90° 时，cos θ = 0，故向量正交。若 θ = 0° 或 180°，两向量平行，此时点积等于 ±|a||b|。
In IB exam problems you are often asked to find acute or obtuse angles between lines, between a line and a plane, or between two planes using vector methods.
在 IB 考试中，常要求你利用向量方法求直线间、直线与平面间或两平面间的锐角或钝角。
9. Vector Equation of a Line | 直线的向量方程
A line in 2D or 3D can be written in vector form as r = a + td, where a is the position vector of a fixed point on the line, d is a direction vector, and t is a scalar parameter.
二维或三维中的直线可用向量形式表示为 r = a + td，其中 a 是直线上一定点的位置向量，d 为方向向量，t 为标量参数。
t d₂, z = a₃ + t d₃. Eliminating t yields the Cartesian (symmetric) form (when possible).
将对应分量等式写出即得参数方程：x = a₁ + t d₁, y = a₂ + t d₂, z = a₃ + t d₃。消去 t 可得标准（对称）形式（视情况而定）。
Two lines are parallel if their direction vectors are scalar multiples. They intersect if solving simultaneous equations yields a consistent value of t. The angle between two lines is the angle between their direction vectors.
若两直线的方向向量成标量倍数关系，则它们平行。若联立方程组得出一致的参数 t 值，则两直线相交。两直线间的夹角即它们方向向量间的夹角。
10. Vector Equation of a Plane | 平面的向量方程 (HL)
A plane can be described by the vector equation r · n = a · n, where n is a normal vector perpendicular to the plane and a is the position vector of a known point on the plane.
平面可用向量方程 r · n = a · n 描述，其中 n 是垂直于该平面的法向量，a 是平面上已知点的位置向量。
Alternatively, a plane can be expressed as r = a + λd₁ + μd₂, using two non‑parallel direction vectors lying in the plane. This parametric form is useful for finding intersections with lines.
平面也可写为 r = a + λd₁ + μd₂，其中 d₁ 和 d₂ 为平面内两个不平行的方向向量。这种参数形式有助于求直线与平面的交点。
The scalar product form r · n d corresponds to the Cartesian equation ax + by + cz = d, which reveals the normal vector n = (a, b, c) immediately.
数量积形式 r · n = d 对应笛卡尔方程 ax + by + cz = d，从中可直接读出法向量 n = (a, b, c)。
11. Cross Product (Vector Product) | 叉积（向量积）(HL)
The cross product of two 3D vectors a and b is a vector a × b perpendicular to both, with magnitude |a||b| sin θ. The direction follows the right‑hand rule.
两个三维向量 a 与 b 的叉积是一个同时垂直于二者的向量 a × b，其大小为 |a||b| sin θ，方向遵循右手定则。
Using components, the cross product is computed via a 3×3 determinant: a × b = (a₂b₃ - a₃b₂)i + (a₃b₁ - a₁b₃)j + (a₁b₂ - a₂b₁)k.
利用分量，叉积可通过 3×3 行列式计算：a × b = (a₂b₃ - a₃b₂)i + (a₃b₁ - a₁b₃)j + (a₁b₂ - a₂b₁)k。
Key properties: a × b is anti‑commutative (a × b = -b × a), and its magnitude equals the area of the parallelogram spanned by a and b.
重要性质：叉积反对称（a × b = -b × a），其大小等于以 a 和 b 为邻边的平行四边形的面积。
12. Applications: Distances and Areas | 应用：距离与面积
n the line to the external point onto the perpendicular direction. A common formula is d = |(p - a) × d| / |d|, where p is the point, a is on the line, and d is the direction vector.
点到直线的最短距离可通过将直线上一点到外部点的向量向垂直方向投影求得。常用公式为 d = |(p - a) × d| / |d|，其中 p 为给定点，a 在直线上，d 为方向向量。
The distance from a point to a plane r · n = d is given by |(p · n - d)| / |n|, where p is the point's position vector. This is derived by projecting onto the unit normal.
点到平面 r · n = d 的距离为 |(p · n - d)| / |n|，其中 p 是点的位置向量。该结果通过投影到单位法向量上得出。
The area of a triangle with sides represented by vectors a and b is ½|a × b|. The volume of a parallelepiped defined by three vectors a, b, c is |(a × b) · c|, the scalar triple product.
以向量 a 和 b 为两边的三角形面积为 ½|a × b|。由三个向量 a、b、c 定义的平行六面体的体积为 |(a × b) · c|，即标量三重积。
These vector techniques provide elegant methods for solving 3D geometry problems without relying heavily on coordinate geometry.
这些向量技巧为求解三维几何问题提供了优雅的方法，而无需过多依赖坐标几何。
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