2.9
矢量积（叉积）
两个矢量的矢量乘法会产生一个矢量积，其大小等于各个矢量的乘积乘以两个矢量与垂直于两个矢量的方向之间夹角的正弦值。 由于总是有两个方向垂直于给定平面，每侧一个，因此矢量积的方向由右手定则确定。
现在考虑两个矢量的叉积。 想象一下围绕垂直线旋转第一个矢量，直到它与第二个矢量对齐，选择两个矢量之间较小的夹…
两个向量的叉积或矢量积是它们的模的乘积与夹角正弦的乘积，其方向垂直于这两个向量所在的平面。
从几何上看，它是向量张成的平行四边形的面积。
根据右手定则，若右手的食指和中指分别指向第一个和第二个矢量， 则拇指指向它们的叉积方向。
第二个向量与第一个向量的叉乘方向相反。因此，叉乘的顺序很重要。
门的旋转程度由力矩决定，力矩是铰链与施力点之间的径向距离与所施加的力的矢量叉积。
如果作用力沿着径向距离的方向或相反方向作用，则门不会发生转动。
对于坐标轴的单位矢量，其与自身的叉乘结果为零。而与其他坐标轴单位矢量的叉乘结果，则为按循环顺序排列的第三个单位矢量。
向量 A 和 B 的叉乘结果的分量分别为 AyBz 减去 AzBy、AzBx 减去 AxBz，以及 AxBy 减去 AyBx。
View the full transcript and gain access to JoVE Core videos
Q1: What is the cross product and how is it calculated geometrically?
The cross product of two vectors equals the product of their magnitudes and the sine of the angle between them, directed perpendicular to their plane. Geometrically, it represents the area of the parallelogram spanned by the two vectors. The magnitude depends on both vector sizes and their angular relationship, while direction follows the right-hand rule.
Q2: How does the right-hand rule determine cross product direction?
Point your right hand's index and middle fingers along the first and second vectors respectively. Your thumb then points in the cross product direction. If you reverse the vector order, the thumb points opposite, showing that cross products are anti-commutative. This rule resolves the ambiguity of two perpendicular directions.
Q3: Why does the order of vectors matter in cross product calculations?
The cross product is anti-commutative: reversing vector order reverses the result's direction. The cross product of vector B with vector A points opposite to the cross product of A with B. This directional reversal means A × B ≠ B × A, making order critical for correct physical interpretation.
Q4: What are the component formulas for calculating a cross product?
For vectors A and B, the cross product components are: (AyBz − AzBy), (AzBx − AxBz), and (AxBy − AyBx). These formulas use vector components in the Cartesian coordinate system to compute each component of the resulting cross product vector systematically.
Q5: When is the cross product of two vectors equal to zero?
The cross product equals zero when vectors are parallel or antiparallel, including when a vector is crossed with itself. Since the sine of 0° and 180° is zero, any vector pair aligned in the same or opposite direction produces zero cross product, regardless of magnitude.
Q6: How does torque relate to the cross product in physics?
Torque is the cross product of the radial distance vector from a hinge with the applied force vector. A door rotates only when force acts perpendicular to the radial distance; force along or opposite the radial direction produces zero torque. This demonstrates how cross product direction determines rotational effect.
Q7: What is the cross product of unit vectors in cyclic order?
Unit vectors of perpendicular axes follow a cyclic pattern: each unit vector crossed with itself yields zero, but crossed with another unit vector yields the third one in cyclic order. This property simplifies calculations and reflects the fundamental orthogonal structure of three-dimensional space.