📚 Properties of the Vector Product | 向量积的性质
The vector product a × b is defined only for vectors in three-dimensional space. It returns a vector, not a scalar, and its properties drive calculations in geometry, mechanics and further pure mathematics. This article collects the essential properties of the vector product, with component checks, geometric meaning and exam-style advice, for AQA A-Level Mathematics.
向量积 a × b 只对三维空间中的向量有定义。它给出的是向量而不是标量，其性质是几何、力学与进阶纯数学中各种计算的基础。本文汇总向量积的核心性质，配合分量验证、几何意义与应试建议，面向 AQA A-Level 数学。
1. Definition and Component Form | 定义与分量形式
For vectors a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃), the vector product a × b is defined by the component formula below. The result is a vector in R³, so every calculation ends with three numbers, never one.
对向量 a = (a₁, a₂, a₃) 与 b = (b₁, b₂, b₃)，向量积 a × b 由下面的分量公式定义。结果是 R³ 中的一个向量，所以每次计算最后得到的是三个数，而不是一个数。
a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁)
The first component uses only the second and third components of the original vectors, the second component uses the first and third, and the third component uses the first and second. This cyclic arrangement is far easier to remember in determinant form, expanded along the first row.
第一个分量只用原向量的第二和第三分量，第二个分量用第一和第三分量，第三个分量用第一和第二分量。这种循环排列写成行列式形式后更容易记忆，按第一行展开即可。
| i j k |
| a₁ a₂ a₃ |
| b₁ b₂ b₃ | = i(a₂b₃ − a₃b₂) − j(a₁b₃ − a₃b₁) + k(a₁b₂ − a₂b₁)
If you write the vectors as column vectors, exactly the same three components appear in the answer. This single formula is the foundation for every other property that follows, so it is worth practising until it becomes automatic.
如果把向量写成列向量，答案中出现的仍是同样这三个分量。这一条公式是后续所有性质的基础，值得反复练习到可以熟练写出。
2. Anti-Commutativity | 反交换律
Reversing the order of the two vectors reverses the sign of the result. This is the property known as anti-commutativity, and it is the single most common source of lost marks in vector product questions.
交换两个向量的顺序会使结果变号。这条性质称为反交换律，也是向量积题目中最常见的失分点。
a × b = −(b × a)
The magnitudes are identical, because the magnitude depends only on the lengths and the angle between the vectors, not on the order. Only the direction is reversed, which matches the right-hand rule: swapping the two fingers flips the thumb.
两者的模完全相等，因为模只取决于两个向量的长度和夹角，与顺序无关。改变的只是方向，这与右手定则一致：交换两根手指，拇指方向随之反向。
A quick component check confirms this. Compare the first component of a × b, which is a₂b₃ − a₃b₂, with the first component of b × a, which is b₂a₃ − b₃a₂. They are exact negatives of each other, and the same happens in the second and third components.
用分量快速验证即可确认。比较 a × b 的第一个分量 a₂b₃ − a₃b₂ 与 b × a 的第一个分量 b₂a₃ − b₃a₂，二者互为相反数，第二、第三个分量同样如此。
3. Distributivity over Addition | 对加法的分配律
The vector product distributes over vector addition on either side, provided the order of the factors inside each product is preserved.
向量积对向量加法在左右两侧都满足分配律，但必须保持每个乘积中因子的先后顺序不变。
a × (b + c) = a × b + a × c
(a + b) × c = a × c + b × c
Because of anti-commutativity, the two expansions are not the same expression. In the second line the vector c stays on the right, and it would be wrong to write c × a + c × b. Distributivity is what allows you to expand brackets such as (a + 2b) × c one term at a time.
由于存在反交换律，这两个展开式并不相同。第二行中向量 c 始终在右侧，写成 c × a + c × b 就是错误的。正是分配律让我们可以逐项展开像 (a + 2b) × c 这样的括号。
- Expand scalars first: (2a) × (3b) = 6(a × b).
- 先提出数字系数：(2a) × (3b) = 6(a × b)。
- Then expand brackets term by term, keeping the order fixed.
- 再逐项展开括号，保持顺序不变。
- Cancel any term of the form v × v, since it equals the zero vector.
- 形如 v × v 的项可直接消去，因为它等于零向量。
4. Multiplication by a Scalar | 数乘性质
Scalars can be moved freely in and out of a vector product, in either position.
标量可以自由地在向量积内外移动，放在哪一侧都可以。
(λa) × b = λ(a × b) = a × (λb)
Here λ is any real number, positive or negative. This property means a vector product is linear in each argument separately, which is exactly what makes bracket expansion possible.
这里 λ 是任意实数，可正可负。该性质说明向量积对每一个自变量都是线性的，这正是括号可以展开的原因。
Do not confuse this with associativity. The expression a × (b × c) is perfectly meaningful, but it is generally not equal to (a × b) × c. The vector product is not associative, so the position of brackets matters and must never be dropped.
不要把这条性质与结合律混为一谈。表达式 a × (b × c) 本身是有意义的，但它一般不等于 (a × b) × c。向量积不满足结合律，因此括号的位置很重要，绝不能随意省略。
5. The Zero Vector and Parallel Vectors | 零向量与平行向量
The vector product of two non-zero vectors is the zero vector if and only if the two vectors are parallel. This is the decisive test for parallelism, and it works in both directions.
两个非零向量的向量积为零向量，当且仅当这两个向量平行。这是判断平行最有效的检验方法，正反两个方向都成立。
a × b = 0 ⇔ a = 0 or b = 0 or a ∥ b
In particular a × a = 0 for every vector a, because a vector is trivially parallel to itself. Likewise i × i = j × j = k × k = 0.
特别地，对任意向量 a 都有 a × a = 0，因为向量与自己显然平行。同理 i × i = j × j = k × k = 0。
ollapses to zero. A vector of length zero has no direction, so the zero vector is the only possible answer.
原因是正弦公式。若两向量平行，夹角 θ 为 0° 或 180°，于是 sin θ = 0，模随之收缩为零。长度为零的向量没有方向，所以零向量是唯一可能的答案。
6. Magnitude and the Sine Formula | 模长与正弦公式
The magnitude of the vector product of two non-zero vectors equals the product of their magnitudes multiplied by the sine of the angle between them.
两个非零向量的向量积的模，等于两向量模的乘积再乘以它们夹角的正弦。
|a × b| = |a| |b| sin θ, 0° ≤ θ ≤ 180°
Because sin θ is never negative on this interval, the magnitude formula always returns a non-negative value. There are two practical routes to this quantity: compute the cross product first and then take its modulus, or use the dot product to find cos θ and then convert to sin θ.
由于 sin θ 在该区间上恒非负，模长公式给出的结果永远是非负值。实际计算有两条路径：先算出向量积再取模；或者用点积求出 cos θ，再换算成 sin θ。
A useful identity links the two products and avoids computing the angle altogether.
有一个很有用的恒等式可以把两个乘积联系起来，从而完全不必求夹角。
|a × b|² = |a|²|b|² − (a · b)²
It follows directly from the Pythagorean identity sin²θ + cos²θ = 1 multiplied through by |a|²|b|², and it is often the fastest route in an exam when both products are needed anyway.
它由恒等式 sin²θ + cos²θ = 1 两边同乘 |a|²|b|² 直接得到，在考试中往往是最快的路径，尤其是题目本来就要求两个乘积的时候。
7. Direction and the Right-Hand Rule | 方向与右手定则
The vector a × b is perpendicular to both a and b. This single fact is the reason the vector product is used to find normals to planes.
向量 a × b 同时垂直于 a 与 b。正是这一事实使得向量积可以用来求平面的法向量。
(a × b) · a = 0 and (a × b) · b = 0
The direction is fixed by the right-hand rule. Point the fingers of your right hand along a, curl them towards b, and your thumb indicates the direction of a × b. Reversing the order of the fingers reverses the thumb, which is anti-commutativity made visible.
方向由右手定则确定。把右手指沿 a 的方向伸出，向 b 的方向弯曲，拇指所指即 a × b 的方向。交换手指顺序，拇指随之反向，这就是反交换律的直观体现。
Because a perpendicular direction can be read from two opposite sides, the order of the factors is part of the answer. If a question asks for a normal vector, either ordering is acceptable, but if it asks for the normal in a particular sense, the order matters.
由于垂直方向可以从两个相反的侧面读出，因子的先后顺序本身就是答案的一部分。若题目只要求一个法向量，两种顺序都可以；若要求特定指向的法向量，顺序就不能随意调换。
8. Unit Vectors and Cyclic Relations | 单位向量与循环关系
The standard basis vectors i, j, k satisfy a neat cyclic pattern that is worth memorising, because it lets you expand products mentally.
标准基向量 i、j、k 满足一个整洁的循环规律，值得记牢，因为它可以让你在脑中直接展开乘积。
i × j = k, j × k = i, k × i = j
wards follows the cycle i → j → k → i. Reading any pair backwards introduces a minus sign.
顺着上面的顺序读，构成循环 i → j → k → i。任何一对反过来写都会多出一个负号。
j × i = −k, k × j = −i, i × k = −j
| × | i | j | k |
| i | 0 | k | −j |
| j | −k | 0 | i |
| k | j | −i | 0 |
Using the table, a product such as (2i + 3j) × (i − k) expands to 2i × i − 2i × k + 3j × i − 3j × k = 0 + 2j − 3k − 3i, that is (−3, 2, −3).
利用这张表，(2i + 3j) × (i − k) 可展开为 2i × i − 2i × k + 3j × i − 3j × k = 0 + 2j − 3k − 3i，即 (−3, 2, −3)。
9. The Scalar Triple Product and Volume | 混合积与体积
Combining the dot product with the vector product gives the scalar triple product, a single number with a strong geometric meaning.
把点积与向量积结合起来就得到混合积，它是一个具有明确几何意义的数值。
a · (b × c) = |a| |b × c| cos φ
The absolute value of this scalar equals the volume of the parallelepiped formed by the three vectors, and the signed value tells you whether a, b, c form a right-handed or left-handed set.
该数值的绝对值等于以这三个向量为棱的平行六面体的体积，而带符号的值则说明 a、b、c 构成右手系还是左手系。
- If a · (b × c) = 0, the three vectors are coplanar, so the volume collapses to zero.
- 若 a · (b × c) = 0，则三向量共面，体积收缩为零。
- Cyclic permutations leave the value unchanged: a · (b × c) = b · (c × a) = c · (a × b).
- 循环置换不改变数值：a · (b × c) = b · (c × a) = c · (a × b)。
- Swapping any two vectors changes the sign, so the triple product is anti-symmetric.
- 交换任意两个向量会改变符号，因此混合积是反对称的。
10. Areas of Triangles and Parallelograms | 三角形与平行四边形的面积
Because |a × b| anned by the two vectors. Halving it gives the area of the triangle on the same two sides.
由于 |a × b| 等于底乘高，它表示以这两个向量为邻边的平行四边形的面积。取一半即得以这两边构成的三角形面积。
A(parallelogram) = |a × b|, A(triangle) = ½ |a × b|
In coordinate geometry, if a triangle has vertices P, Q and R, the area is half the modulus of PQ × PR. The same idea gives the perpendicular distance from a point P to a line through A with direction d.
在坐标几何中，若三角形顶点为 P、Q、R，其面积等于 PQ × PR 模的一半。同样的思路还能给出点 P 到过 A 点、方向为 d 的直线的垂直距离。
distance = |AP × d| ÷ |d|
These two results are the standard way to finish off vector geometry questions, and they often appear together with a request to find a unit normal by dividing the cross product by its modulus.
这两个结果是收尾向量几何题的常规手段，而且常与"用向量积除以它的模求单位法向量"的要求一起出现。
11. Worked Example | 典型例题
Take a = (1, 2, 3) and b
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