📚 Dot (scalar) product | 点积（标量积）
The dot product, also called the scalar product, is a fundamental operation in vector algebra that combines two vectors to produce a single scalar quantity. It plays a key role in determining angles, lengths, projections, and orthogonality, and is widely used in geometry, physics, and engineering. In the IB Mathematics curriculum, particularly at Higher Level, you are expected to compute dot products in both two and three dimensions, apply their properties, and interpret the result geometrically.
点积，又称标量积，是向量代数中的基本运算，它将两个向量结合得到一个标量。点积在确定夹角、长度、投影和正交性方面起着关键作用，广泛应用于几何、物理和工程学。在IB数学课程中，特别是高水平，你需要掌握在二维和三维空间中计算点积，运用其性质，并从几何角度解释结果。
1. Geometric definition of the dot product | 点积的几何定义
The geometric definition of the dot product relates directly to the magnitudes of the two vectors and the angle between them. For two vectors a and b, the dot product is given by a · b = |a| |b| cos θ, where |a| and |b| denote the lengths (magnitudes) of the vectors, and θ is the angle between them when their tails are placed at the same point. The angle θ is always taken to be between 0 and π radians (0° and 180°).
点积的几何定义直接与两个向量的模长及它们的夹角相关。对于两个向量 a 和 b，点积定义为 a · b = |a| |b| cos θ，其中 |a| 和 |b| 表示向量的模长，θ 是它们起点重合时形成的夹角。夹角 θ 的取值范围总是 0 到 π（即 0° 到 180°）。
a · b = |a| |b| cos θ
egative when θ is obtuse (cos θ < 0). It captures the idea of how much one vector extends in the direction of another.
这一定义表明，当 θ 为锐角时点积为正，当 θ = 90° 时点积为零，当 θ 为钝角时点积为负。它体现了其中一个向量在另一个向量方向上延伸的程度。
2. Algebraic (component) definition | 代数（坐标）定义
When vectors are expressed in component form relative to Cartesian axes, the dot product can be computed by summing the products of corresponding components. In two dimensions, for a = (a₁, a₂) and b = (b₁, b₂), we have a · b = a₁b₁ + a₂b₂. In three dimensions, for a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃), it extends to a · b definition is extremely useful for manual calculation and forms the basis for many proofs.
当向量以直角坐标系中的分量形式给出时，点积可以通过对应分量乘积之和来计算。在二维中，对于 a = (a₁, a₂) 和 b = (b₁, b₂)，有 a · b = a₁b₁ + a₂b₂。在三维中，对于 a = (a₁, a₂, a₃) 和 b = (b₁, b₂, b₃)，扩展为 a · b = a₁b₁ + a₂b₂ + a₃b₃。这一代数定义非常便于计算，也是许多证明的基础。
a · b = a₁b₁ + a₂b₂ (2D) or a₁b₁ + a₂b₂ + a₃b₃ (3D)
It is important to remember that the result is a scalar, not a vector. Both the geometric and algebraic definitions are equivalent, and the link between them can be established using the cosine rule. This equivalence allows us to switch flexibly between geometric insight and algebraic computation.
必须记住，结果是一个标量，而不是向量。几何定义和代数定义是等价的，二者之间的联系可以通过余弦定理建立。这种等价使得我们能够在几何直观与代数计算之间灵活切换。
3. Calculating the dot product in 2D and 3D | 二维和三维点积的计算
Let’s work through practical examples. In two dimensions, take u = (3, -2) and v = (1, 4). Their dot product is u · v = 3×1 + (-2)×4 = 3 - 8 = -5. In three dimensions, for p = (2, 0, -3) and q = (-1, 4, 2), we have p · q 6 = -8. Notice that a negative dot product indicates an obtuse angle between the vectors.
我们来做些实际例子。在二维空间中，取 u = (3, -2) 和 v = (1, 4)，其点积为 u · v = 3×1 + (-2)×4 = 3 - 8 = -5。在三维空间中，对于 p = (2, 0, -3) 和 q = (-1, 4, 2)，有 p · q = 2×(-1) + 0×4 + (-3)×2 = -2 + 0 - 6 = -8。注意负的点积表示向量间的夹角为钝角。
To check your work, you can always calculate the magnitudes and find the angle to see if the algebraic result makes sense geometrically. For instance, |u| = √9+4 = √13, |v| = √1+16 = √17; since the product is negative, cos θ must be negative, confirming an obtuse angle.
为了检验结果，你可以计算模长并求出夹角，从几何角度确认代数结果是否合理。例如 |u| = √9+4 = √13，|v| = √1+16 = √17；由于点积为负，cos θ 必为负值，验证了夹角为钝角。
4. Properties of the dot product | 点积的性质
The dot product obeys several algebraic rules that make it a powerful tool. First, it is commutative: a · b = b · a. Second, it is distributive over vector addition: a · (b + c) = a · b + a · c. Third, it interacts with scalar multiplication as follows: (k a) · b = k (a · b) = a · (k b). Fourth, the dot product of a vector with itself gives the square of its magnitude: a · a = |a|².
点积满足多项代数规则，这使得它成为一个强大的工具。首先，它满足交换律：a · b = b · a。其次，它对向量加法满足分配律：a · (b + c) = a · b + a · c。第三，它与标量乘法的关系为：(k a) · b = k (a · b) = a · (k b)。第四，向量与自身的点积等于其模长的平方：a · a = |a|²。
These properties closely resemble ordinary multiplication of real numbers, but it is crucial to remember that the dot product is not associative; an expression like (a · b) c is not defined because you cannot form a dot product between a scalar and a vector. The distributive law can be applied repeatedly to expand brackets involving dot products, much like expanding algebraic expressions.
这些性质与实数的普通乘法非常相似，但要记住点积不满足结合律；像 (a · b) c 这样的表达式是没有定义的，因为你不能让标量与向量作点积。分配律可以重复应用来展开包含点积的括号，与代数式的展开非常类似。
| Property | Rule |
|---|---|
| Commutative | a · b = b · a |
| Distributive | a · (b + c) = a · b + a · c |
| Scalar multiplication | (ka) · b = k(a · b) |
| Self dot product | a · a = |a|² |
5. Angle between two vectors | 两向量间的夹角
find the angle between two vectors. Rearranging the geometric definition yields cos θ = (a · b) / (|a| |b|). Once the dot product and magnitudes are computed, θ can be found using the inverse cosine function. It is essential to calculate both magnitudes and the dot product accurately to avoid sign errors.
点积最常见的应用之一是求两个向量之间的夹角。将几何定义变形可得 cos θ = (a · b) / (|a| |b|)。一旦计算出点积和模长，就可以使用反余弦函数求得 θ。必须精确计算模长和点积，以避免符号错误。
cos θ = (a · b) / (|a| |b|)
As an example, let a = (2, -1, 2) and b = (-2, 0, 1). Then a · b = 2×(-2) + (-1)×0 + 2×1 = -4 + 0 + 2 = -2. |a| = √4 + 1 + 4 = √9 = 3, |b| = √4 + 0 + 1 = √5. Thus cos θ = -2 / (3√5 ng θ ≈ 107.4°. The obtuse angle tells us the vectors point in roughly opposite directions.
例如，设 a = (2, -1, 2) 和 b = (-2, 0, 1)。则 a · b = 2×(-2) + (-1)×0 + 2×1 = -4 + 0 + 2 = -2。 |a| = √4+1+4 = 3，|b| = √4+0+1 = √5。因此 cos θ = -2/(3√5) ≈ -0.298，得 θ ≈ 107.4°。钝角表明这两个向量大致指向相反的方向。
6. Orthogonal and perpendicular vectors | 正交向量与垂直向量
Vectors a and b are orthogonal (perpendicular) if and only if their dot product is zero. For non-zero vectors, a · b = 0 ⇔ or perpendicularity. The zero vector is technically orthogonal to every vector because 0 · b e usually require both vectors to be non-zero to speak of perpendicular lines or segments.
两个向量 a 和 b 正交（垂直）当且仅当它们的点积为零。对于非零向量，a · b = 0 ⇔ θ = 90°。这提供了一个简单的代数方法来检验垂直性。零向量在技术上与任何向量正交，因为 0 · b = 0，但在几何情境中，我们通常要求两个向量都是非零的才能说直线或线段垂直。
For example, the vectors u = (3, 6) and v = -12 + 12 = 0, so they are perpendicular. In three dimensions, the standard basis vectors i, j, k are mutually orthogonal, since i · j = j · k = k · i onal pairs of direction vectors is crucial when working with planes and lines in 3D space.
例如，向量 u = (3, 6) 和 v = (-4, 2) 的点积为 3×(-4) + 6×2 = -12 + 12 = 0，因此它们垂直。在三维空间中，标准基向量 i、j、k 两两正交，因为 i · j = j · k = k · i = 0。在处理三维空间中的平面和直线时，识别方向向量的正交对至关重要。
7. Scalar projection of one vector onto another | 向量在另一个向量上的标量投影
The scalar projection (or component) of vector a onto vector b is the signed length of the "shadow" of a along the direction of b. It is denoted by compb a and calculated as compb a = (a · b) / |b|. The sign of the scalar projection indicates whether the projection points in the same direction as b (positive) or opposite (negative). Its absolute value equals |a| cos θ, the length of the adjacent side in the right triangle formed by the projection.
向量 a 在向量 b 上的标量投影（或分量）是 a 沿 b 方向的“影子”的有向长度。记为 compb a，计算公式为 compb a = (a · b) / |b|。标量投影的符号表明投影方向与 b 相同（正号）还是相反（负号）。其绝对值等于 |a| cos θ，即投影构成的直角三角形中的邻边长度。
compb a = (a · b) / |b|
Suppose a = (5, 1) and b = (2, 2). Then a · b = 10 + 2 = 12, |b| = √4+4 = √8 = 2√2. The scalar projection is 12 / (2√2) = 6/√2 = 3√2 ≈ 4.24. This means that if we follow the direction of b, the component of a in that direction has length about 4.24.
例如，设 a = (5, 1) 和 b = (2, 2)。则 a · b = 10+2=12，|b| = √4+4=√8=2√2。标量投影为 12/(2√2) = 6/√2 = 3√2 ≈ 4.24。这意味着沿着 b 的方向，a 在该方向上的分量长度约为 4.24。
8. Vector projection of one vector onto another | 向量在另一个向量上的向量投影
While the scalar projection gives a signed length, the vector projection of a onto b is the actual vector parallel to b that represents the projection. It is obtained by multiplying the scalar projection by the unit vector in the direction of b, i.e., proj b a = ((a · b) / |b|²) b. This vector has magnitude equal to the absolute value of the scalar projection.
标量投影给出有向长度，而向量 a 在 b 上的向量投影则是平行于 b 的实际投影向量。它通过将标量投影乘以 b 方向的单位向量得到，即 proj b a = ((a · b) / |b|²) b。该向量的模等于标量投影的绝对值。
projb a = (a · b / |b|²) b
Using the previous example, a = (5, 1), b = (2, 2). The scalar factor is (a · b) / |b|² = 12 / 8 = 1.5. So proj b a = 1.5 × (2, 2) = (3, 3). Notice that the projection vector points in the same direction as b because the scalar projection is positive. Vector projections are particularly useful in physics to resolve forces into components.
沿用上例，a = (5, 1)，b = (2, 2)。标量因子为 (a · b) / |b|² = 12/8 = 1.5。所以 proj b a = 1.5 × (2, 2) = (3, 3)。注意投影向量与 b 同方向，因为标量投影为正。向量投影在物理学中分解力时特别有用。
9. Work done by a constant force | 恒力做的功
In physics, the work W done by a constant force F causing a displacement d is defined as the dot product W = F · d. If the force is measured in newtons and displacement in metres, work is measured in joules. The formula highlights that only the component of force in the direction of displacement contributes to work; a force perpendicular to displacement does no work.
在物理学中，恒力 F 产生位移 d 所做的功 W 定义为点积 W = F · d。如果力以牛顿为单位，位移以米为单位，则功的单位是焦耳。该公式表明，只有力在位移方向上的分量才做功；与位移垂直的力不做功。
W = F · d = |F| |d| cos θ
al moves a box 5 m along a horizontal floor. The work done is W = (10)(5) cos 30° = 50 × (√3 ng the box; if the force were vertical, cos 90° = 0, and no horizontal work would be done.
例如，
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