📚 Coplanarity of Vectors | 向量共面性
In AQA A-Level Mathematics, vectors often move from two dimensions into three dimensions. A key idea is coplanarity: deciding whether three vectors, or four points, all lie in the same plane. The main tool is the scalar triple product.
在 AQA A-Level 数学中，向量常从二维扩展到三维。一个核心概念是共面性：判断三个向量或四个点是否位于同一平面。主要工具是标量三重积。
1. What Does Coplanarity Mean? | 什么是共面？
Three vectors are coplanar if they can be placed with the same initial point and all lie in one flat plane. Parallel planes also count because vectors are free to move.
三个向量如果可以平移到同一个起点，并且都位于一个平面上，则称它们共面。由于向量可以自由平移，位于平行平面也算共面。
In 3D, two non-parallel vectors always define a plane, but a third vector might leave that plane. Coplanarity tests whether the third vector stays inside it.
在三维中，两个不平行向量总能确定一个平面，但第三个向量可能离开该平面。共面性检验第三个向量是否仍在该平面内。
2. Vector Notation and Setup | 向量表示与设定
Throughout, vectors are written in bold as a, b, c. In component form, a = (a₁, a₂, a₃), b = (b₁, b₂, b₃), c = (c₁, c₂, c₃).
本文中，向量用粗体 a、b、c 表示。分量形式为 a = (a₁, a₂, a₃)，b = (b₁, b₂, b₃)，c = (c₁, c₂, c₃)。
The dot product gives a scalar, and the cross product gives a vector perpendicular to both original vectors.
点积给出一个标量，叉积给出一个同时垂直于两个原向量的向量。
3. Scalar Triple Product Test | 标量三重积检验
The standard test for coplanarity of a, b, c is:
三个向量 a、b、c 共面的标准检验是：
a · (b × c) = 0
The expression a · (b × c) is called the scalar triple product. If it equals zero, the three vectors are coplanar; if it is nonzero, they are not.
表达式 a · (b × c) 称为标量三重积。若它等于零，则三个向量共面；若不为零，则不共面。
The scalar triple product can be evaluated in any cyclic order:
标量三重积可按任意循环顺序计算：
a · (b × c) = b · (c × a) = c · (a × b)
4. Why the Test Works Geometrically | 该检验的几何原理
The cross product b × c is perpendicular to the plane containing b and c. If a is also in that plane, then the dot product with this perpendicular normal is zero.
叉积 b × c 垂直于 b 和 c 所在平面。如果 a 也在该平面内，那么它与这个垂直法向量的点积为零。
The magnitude |a · (b × c)| gives the volume of the parallelepiped formed by the three vectors. Zero volume means the three vectors are squashed into a plane.
绝对值 |a · (b × c)| 表示三个向量构成的平行六面体体积。体积为零说明三个向量被压扁到一个平面内。
5. Determinant Form for Coordinates | 坐标下的行列式形式
If the vectors are given in components, the scalar triple product is a 3 × 3 determinant:
如果向量以分量给出，标量三重积是一个 3 × 3 行列式：
a · (b × c) = a₁(b₂c₃ − b₃c₂) − a₂(b₁c₃ − b₃c₁) + a₃(b₁c₂ − b₂c₁)
Setting this determinant equal to zero is the quickest coordinate test for coplanarity.
令该行列式等于零是最快的坐标共面性检验方法。
6. Worked Example: Three Vectors | 例题：三个向量
Show that a = (1, 2, 3), b = (4, 5, 6) and c = (7, 8, 9) are coplanar.
证明 a = (1, 2, 3)、b = (4, 5, 6) 和 c = (7, 8, 9) 共面。
First compute the cross product:
先计算叉积：
b × c = (5×9 − 6×8, 6×7 − 4×9, 4×8 − 5×7) = (−3, 6, −3)
Then take the dot product with a:
然后与 a 取点积：
a · (b × c) = 1×(−3) + 2×6 + 3×(−3) = −3 + 12 − 9 = 0
Since the scalar triple product is zero, the three vectors are coplanar.
由于标量三重积为零，这三个向量共面。
7. Coplanarity of Points |
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