Properties of the Vector Product | 向量积的性质

📚 Properties of the Vector Product | 向量积的性质

The vector product, also called the cross product, is a fundamental operation in three-dimensional geometry. While the dot product produces a scalar, the vector product produces another vector. Mastering its algebraic properties is essential for solving AQA A-Level problems involving forces, moments, plane normals, and areas.

向量积（又称叉积）是三维几何中的基本运算。点积产生标量，而向量积产生一个新向量。掌握其代数性质对于解决 AQA A-Level 中涉及力、力矩、平面法向量与面积的题目至关重要。

1. A Quick Reminder of the Vector Product | 向量积的快速回顾

For two vectors a and b, the vector product is written a × b. If the angle between the two vectors is θ, measured from a to b, then the magnitude is |a||b| sin θ.

对于两个向量 a 和 b，向量积记为 a × b。若两向量之间的夹角为 θ（从 a 到 b 方向度量），则其模长为 |a||b| sin θ。

a × b = |a||b| sin θ n̂

where n̂ is a unit vector perpendicular to both a and b, with direction given by the right-hand rule. In component form, for a = a₁i + a₂j + a₃k and b = b₁i + b₂j + b₃k:

其中 n̂ 是同时垂直于 a 和 b 的单位向量，方向由右手定则确定。在分量形式中，设 a = a₁i + a₂j + a₃k，b = b₁i + b₂j + b₃k：

a × b = (a₂b₃ − a₃b₂)i + (a₃b₁ − a₁b₃)j + (a₁b₂ − a₂b₁)k

This expression is often evaluated most conveniently using a 3 × 3 determinant or the minor-cofactor expansion.

这个表达式通常利用 3 × 3 行列式或余子式展开来计算最为方便。

2. Anti-Commutativity | 反交换律

The most distinctive algebraic property of the vector product is that it is anti-commutative. Reversing the order of the factors changes the sign:

向量积最独特的代数性质是反交换律。交换因子的顺序会改变符号：

a × b = −(b × a)

This happens because reversing the order flips the direction of the unit normal n̂ according to the right-hand rule. In contrast, the dot product is commutative, since a · b = b · a.

这是因为顺序颠倒时，右手定则使得单位法向量 n̂ 的方向反转。相比之下，点积满足交换律，因为 a · b = b · a。

Anti-commutativity produces the useful unit-vector results: i × j = k, j × k = i, k × i = j, while j × i = −k, k × j = −i, and i × k = −j.

反交换律给出了有用的单位向量结果：i × j = k，j × k = i，k × i = j；而 j × i = −k，k × j = −i，i × k = −j。

3. The Zero Vector Result | 零向量结果

When two vectors are parallel, or identical, the sine of the angle between them is zero, so the vector product vanishes:

当两个向量平行或相同时，两向量夹角的正弦为零，因此向量积为零：

a × a = 0

More generally, a × b = 0 if and only if a

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