📚 The Dot Product (Scalar Product) of Vectors | 向量的点积（标量积）
The dot product, also known as the scalar product, is one of the most important operations in vector algebra. In IB Mathematics, it provides a bridge between algebra and geometry, enabling us to calculate angles, determine perpendicularity, and solve real-world problems involving projections and work.
点积（又称标量积）是向量代数中最重要的运算之一。在IB数学中，它连接了代数与几何，使我们能够计算夹角、判断垂直关系，并解决涉及投影和做功等现实问题。
1. Definition of the Dot Product | 点积的定义
The dot product of two vectors a and b is a scalar quantity defined as:
两个向量 a 与 b 的点积是一个标量，定义为：
a · b = |a||b| cos θ
where θ is the angle between the two vectors when placed tail-to-tail.
其中 θ 是两个向量起点重合时的夹角。
In coordinate form, for vectors in 2D and 3D, the dot product is computed by multiplying corresponding components and summing the results:
在坐标形式下，对于二维和三维向量，点积通过将对应分量相乘后相加得到：
a · b = a₁b₁ + a₂b₂ + a₃b₃
For two-dimensional vectors, the third term is omitted. The result is always a real number, not a vector.
对于二维向量，省略第三项。结果总是一个实数，而不是向量。
2. Geometric Interpretation | 几何意义
Geometrically, the dot product measures how much one vector extends in the direction of another. If you project vector b onto a, the length of that projection is |b| cos θ. Multiplying by |a| gives a · b. Therefore the dot product is the product of the length of one vector and the scalar projection of the other onto it.
从几何上看，点积衡量一个向量在另一个向量方向上的“伸展”程度。如果将向量 b 投影到 a 上，投影长度为 |b| cos θ，再乘以 |a| 即得 a · b。因此，点积是一个向量的长度与另一个向量在其方向上的标量投影之积。
The sign of the dot product reveals the relative direction of the vectors: positive when the angle is acute (0° < θ < 90°), zero when θ = 90°, and negative when the angle is obtuse (90° < θ < 180°).
点积的正负可以反映两向量的方向关系：夹角为锐角（0° < θ < 90°）时为正，θ = 90° 时为零，夹角为钝角（90° < θ < 180°）时为负。
3. Algebraic Properties | 代数性质
For any vectors a, b, c and any scalar k, the following properties hold:
对于任意向量 a、b、c 及任意标量 k，下列性质成立：
Commutative: a · b = b · a
交换律：a · b = b · a
Distributive over addition: a · (b + c) = a · b + a · c
加法分配律：a · (b + c) = a · b + a · c
Associative with scalar multiplication: (k a) · b = k (a · b)
与数乘的结合律：(k a) · b = k (a · b)
4. Computing the Dot Product in 2D and 3D | 二维与三维中的点积计算
Let us compute a concrete example. Given a = (3, -2, 4) and b = (1, 5, -2), the dot product is:
让我们计算一个具体例子。已知 a = (3, -2, 4)，b = (1, 5, -2)，则点积为：
a · b = 3×1 + (-2)×5 + 4×(-2) = 3 - 10 - 8 = -15
The negative result tells us that the angle between a and b is obtuse.
结果为负，说明 a 与 b 之间的夹角是钝角。
For 2D vectors, such as p = (4, -1) and q = (2, 7), we have:
对于二维向量，如 p = (4, -1)，q = (2, 7)，有：
p · q = 4×2 + (-1)×7 = 1
5. Angle between Two Vectors | 两向量的夹角
By rearranging the definition, we can find the angle between two vectors:
通过变形定义，我们可以求出两向量之间的夹角：
cos θ = (a · b) / (|a||b|)
where θ is measured in the interval [0°, 180°].
其中 θ 在区间 [0°, 180°] 内取值。
For example, let a = (2, 1) and b = (1, 3). Then a · b = 2×1 + 1×3 = 5, |a| = √2² + 1² = √5, and |b| = √1² + 3² = √10. Hence:
例如，设 a = (2, 1)，b = (1, 3)。则 a · b = 2×1 + 1×3 = 5，|a| = √2² + 1² = √5，|b| = √1² + 3² = √10。因此：
cos θ = 5 / (√5 × √10) = 1/√2, so θ = 45°.
cos θ = 5 / (√5 × √10) = 1/√2，所以 θ = 45°。
6. Perpendicular and Parallel Vectors | 垂直与平行向量
Two non-zero vectors are perpendicular if and only if their dot product is zero:
两个非零向量垂直当且仅当它们的点积为零：
a ⊥ b ⇔ a · b = 0
This is a powerful test in IB questions, especially when checking whether lines or planes are perpendicular.
这是IB考试中非常有力的判定方法，尤其在判断直线或平面是否垂直时。
Vectors are parallel when one is a scalar multiple of the other, which also implies that the angle between them is 0° or 180°. The dot product can be used to confirm this, since for parallel vectors cos θ = ±1, so a · b = ±|a||b|.
向量平行意味着其中一个向量是另一个的标量倍数，其夹角为 0° 或 180°。点积可用于验证平行关系，因为平行向量满足 cos θ = ±1，故 a · b = ±|a||b|。
7. Vector Projection | 向量投影
The dot product is the key to computing vector projections. The scalar projection of a onto b is:
点积是计算向量投影的关键。a 在 b 方向上的标量投影为：
|a| cos θ = (a · b) / |b|
To find the vector projection, multiply the scalar projection by the unit vector in the direction of b:
要求向量投影，只需将标量投影乘以 b 方向上的单位向量：
p = ((a · b) / |b|²) b
Here p is the projection vector of a onto b. This formula appears frequently in vector geometry, for example when finding the distance from a point to a line.
其中 p 是 a 在 b 方向上的投影向量。这个公式在向量几何中经常出现，例如求点到直线的距离时。
8. Work and Applications in Physics | 做功与物理应用
In physics, the work W done by a constant force F acting through a displacement d is given by the dot product:
在物理中，恒力 F 在位移 d 上所做的功 W 由点积给出：
