📚 The Cross Product of Vectors | 向量的叉积（矢量积）
In 3D geometry, the dot product gives useful scalar information, but many IB problems ask for a perpendicular direction or an oriented area. The cross product, also known as the vector product, is the tool that provides these answers. In this article we examine the definition, right-hand rule, component formula, key properties, and the classical applications that appear in IB Mathematics.
在三维几何中，数量积能提供有用的标量信息，但很多IB问题需要“垂直于给定方向”的向量或有向面积。叉积（也称矢量积）正是解决这类问题的工具。本文将系统梳理叉积的定义、右手定则、分量公式、重要性质以及IB数学中的经典应用。
1. Why We Need the Cross Product | 为什么需要叉积
The dot product returns a scalar, so it cannot describe direction. In 3D problems we often need to find a vector that is perpendicular to two given vectors, for example a normal to a plane or an area vector in electromagnetism. The cross product fills this gap by combining two vectors into a third vector perpendicular to both.
点积返回的是一个标量，因此无法描述方向。在三维问题中，我们经常需要找到一个与两个已知向量都垂直的向量，例如平面的法向量或电磁学中的面积矢量。叉积恰好填补了这一空白：它将两个向量组合成第三个与二者都垂直的向量。
2. Geometric Definition of the Cross Product | 叉积的几何定义
For two non-zero vectors a and b, let θ be the angle between them, where 0 ≤ θ ≤ π. Their cross product is a new vector whose magnitude is |a||b| sin θ, and whose direction is perpendicular to both a and b. The unit vector n̂ in that direction is chosen by the right-hand rule.
设两个非零向量 a 和 b 的夹角为 θ，且 0 ≤ θ ≤ π。它们的叉积是一个新向量，其大小为 |a||b| sin θ，方向垂直于 a 和 b 所确定的平面。该方向的单位向量 n̂ 由右手定则确定。
a × b = |a||b| sin θ n̂
If a and b are parallel, then sin θ = 0 and the cross product is the zero vector. The magnitude |a × b| also gives the area of the parallelogram formed by a and b.
如果 a 与 b 平行，则 sin θ = 0，叉积为零向量。同时，|a × b| 等于以 a 和 b 为邻边所构成平行四边形的面积。
3. The Right-Hand Rule | 右手定则
To determine the direction of a × b, hold your right hand so that your fingers curl from a toward b through the smaller angle. Your extended thumb then points in the direction of a × b.
为了判断 a × b 的方向，伸开右手，让四指从 a 沿较小夹角转向 b，此时大拇指所指方向就是 a × b 的方向。
If the order is reversed, the thumb points in the opposite direction. Hence a × b = −(b × a). This is why the cross product is called anti-commutative.
如果交换顺序，大拇指将指向相反方向，因此 a × b = −(b × a)。这就是叉积被称为反交换律运算的原因。
4. Component Form in 3D | 三维分量形式
In Cartesian coordinates, write a = a₁i + a₂j + a₃k and b = b₁i + b₂j + b₃k. Expanding the formal determinant whose first row is i, j, k gives the component formula.
在直角坐标系中，设 a = a₁i + a₂j + a₃k，b = b₁i + b₂j + b₃k。对以 i, j, k 为第一行的形式行列式按第一行展开，就得到分量公式。
a × b = (a₂b₃ − a₃b₂) i − (a₁b₃ − a₃b₁) j + (a₁b₂ − a₂b₁) k
A common mnemonics is to write a × b as the determinant of a 3 × 3 array with first row i, j, k, second row a₁, a₂, a₃, and third row b₁, b₂, b₃, then expand along the first row. Be careful with the minus sign before the j term.
常用的记忆方法是把 a × b 写成一个 3 × 3 行列式的展开：第一行为 i, j, k，第二行为 a₁, a₂, a₃，第三行为 b₁, b₂, b₃，并沿第一行展开。注意 j 项前有负号。
For example, if a = 3i + 2j + k and b = i + j + 2k, then a × b = 3i − 5j + k.
例如，若 a = 3i + 2j + k，b = i + j + 2k，则 a × b = 3i − 5j + k。
5. Properties of the Cross Product | 叉积的性质
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Anti-commutative. a × b = −(b × a).
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