叉积

叉积，即交叉乘积，也被称为向量积、矢量积、或者外积，是指在向量空间中一种对向量的二元运算。与点积不同，它的运算结果是一个伪矢量而不是一个标量，并且得到的矢量与这两个矢量均正交。

叉积的定义

两个矢量 a 和 b 的叉积表示为 a × b (在平时书写时也被写成 a ∧ b ，这样以避免和字母x发生混淆)。叉积可以被定义为下面的形式：

$\mathbf {a} \times \mathbf {b} =\mathbf {\hat {n}} \left|\mathbf {a} \right|\left|\mathbf {b} \right|\sin \theta$

在这里 θ 表示a和 b之间的夹角 (0° ≤ θ ≤ 180°)它位于这两个矢量所定义的平面上。而 n 是一个与a和b均正交的单位矢量。

The problem with this definition is that there are two unit vectors perpendicular to both a and b: if n is perpendicular, then so is −n.

Which vector is the "correct" one by convention depends upon the orientation of the vector space—i.e., on the handedness of the given orthogonal coordinate system (i, j, k). The cross product a × b is defined in such a way that (a, b, a × b) becomes right-handed if (i, j, k) is right-handed, or left-handed if (i, j, k) is left-handed.

An easy way to compute the direction of the resultant vector is the "right-hand rule." If the coordinate system is right-handed, one simply points the forefinger in the direction of the first operand and the middle finger in the direction of the second operand. Then, the resultant vector is coming out of the thumb.

Because the cross product depends on the choice of coordinate system, its result is referred to as a pseudovector. Fortunately, in nature cross products tend to come in pairs, so that the “handedness” of the coordinate system is undone by a second cross product.

The cross product can be represented graphically, with respect to a right-handed coordinate system, as follows:

应用

在物理学光学和计算机图形学中,叉积被用于求物体光照相关问题。

求解光照的核心在于求出物体表面法线，而叉积运算保证了只要已知物体表面的两个非平行矢量(或者不在同一直线的三个点),就可依靠叉积求得法线。

参考

点积