📚 Coplanarity of Vectors | 向量的共面性
In AQA A-level Mathematics, vector questions often require you to decide whether three vectors are coplanar, or whether four given points lie in the same plane. The central idea is the scalar triple product, which tests whether three direction vectors span a solid shape or collapse into a flat plane.
在 AQA A-level 数学中，向量题经常要求你判断三个向量是否共面，或者四个给定点是否位于同一平面内。核心思想是标量三重积，它检验三个方向向量是张成一个立体图形，还是塌缩为一个平面。
Mastering coplanarity helps you solve problems about lines, planes, volumes, and geometric relationships in three dimensions. This article explains the condition, worked examples, and common exam techniques.
掌握共面性有助于你解决三维空间中的直线、平面、体积和几何关系问题。本文介绍判定条件、典型例题和常见考试技巧。
1. What Does Coplanarity Mean? | 什么是共面性？
Three vectors are coplanar if they lie in the same plane. When the vectors are drawn from the same initial point, coplanarity means that a single flat surface can contain all three directed line segments.
三个向量共面是指它们位于同一个平面内。当这些向量从同一起点画出时，共面意味着一个平面可以同时包含所有三条有向线段。
If one vector is a linear combination of the other two, then all three vectors must be coplanar. For example, if c = λa + μb, then c lies in the plane spanned by a and b, so the three vectors cannot form a three-dimensional volume.
如果其中一个向量是另外两个向量的线性组合，那么这三个向量一定共面。例如，若 c = λa + μb，则 c 位于由 a 和 b 张成的平面内，因此这三个向量无法构成三维体积。
This idea is fundamental: coplanarity is equivalent to linear dependence among three vectors in three-dimensional space.
这一思想是根本性的：在三维空间中，共面性等价于三个向量线性相关。
2. Vector Form of a Plane | 平面的向量形式
A plane through the origin is the set of all linear combinations of two non-parallel direction vectors a and b. Any vector in this plane has the form r = λa + μb, where λ and μ are scalar parameters.
通过原点的平面是两个不平行方向向量 a 和 b 的所有线性组合的集合。该平面内的任何向量都具有 r = λa + μb 的形式，其中 λ 和 μ 是标量参数。
For a plane that does not pass through the origin, we introduce a position vector p to a known point on the plane. The general vector equation is r = p + λa + μb.
对于不经过原点的平面，我们引入指向平面上已知点的位置向量 p。一般向量方程为 r = p + λa + μb。
In coplanarity problems, you are often given three direction vectors and asked whether they can all be written inside the same plane. This is exactly the question of whether one vector can be expressed as a linear combination of the other two.
在共面性问题中，题目常常给出三个方向向量，并询问它们是否都能写在同一个平面内。这恰好就是一个向量能否表示为另外两个向量的线性组合的问题。
3. Scalar Triple Product | 标量三重积
The scalar triple product of three vectors a, b, and c is denoted by a · (b × c). It produces a scalar, not a vector, and its absolute value gives the volume of the parallelepiped formed by the three vectors.
三个向量 a、b 和 c 的标量三重积记作 a · (b × c)。它得到一个标量而非向量，其绝对值给出由这三个向量构成的平行六面体的体积。
If the scalar triple product is zero, the volume is
Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图