Coplanarity of Vectors | 向量的共面性

📚 Coplanarity of Vectors | 向量的共面性

In A-Level mathematics, vectors are not only used to describe straight lines and planes, but also to determine whether three or more vectors lie in the same plane. This property is called coplanarity, and it plays a central role in 3D coordinate geometry.

在 A-Level 数学中，向量不仅用于描述直线和平面，还用于判断三个或更多向量是否位于同一平面内。这一性质称为共面性，它在三维坐标几何中扮演着核心角色。

1. What Does Coplanar Mean? | 共面是什么意思？

Three or more vectors are said to be coplanar if they lie in the same plane. In other words, there exists a single plane that contains all of them, provided they are placed with a common initial point.

如果三个或更多向量位于同一平面内，则称它们是共面的。换句话说，当它们从同一起点出发时，存在一个平面能够包含所有这些向量。

For two vectors, coplanarity is always true because any two vectors automatically define a plane (unless they are parallel, in which case infinitely many planes contain them). For three vectors, coplanarity is a non-trivial condition that can be tested algebraically.

对于两个向量，共面性总是成立，因为任意两个向量自然确定一个平面（除非它们平行，此时包含它们的平面有无穷多个）。对于三个向量，共面性是一个非平凡的条件，需要通过代数方法检验。

2. The Scalar Triple Product Test | 标量三重积检验法

The most efficient way to test whether three vectors a, b and c are coplanar is to compute their scalar triple product: a · (b × c).

检验三个向量 a、b 和 c 是否共面最有效的方法是计算它们的标量三重积：a · (b × c)。

If the scalar triple product equals zero, the three vectors are coplanar. If it is non-zero, they are not coplanar.

如果标量三重积等于零，则三个向量共面；如果不等于零，则它们不共面。

a · (b × c) = 0 ⇔ a, b, c are coplanar

This works because b × c is a vector normal to the plane containing b and c. If a also lies in that plane, it must be perpendicular to this normal vector, so the dot product is zero.

其原理在于 b × c 是包含 b 和 c 的平面的法向量。如果 a 也位于该平面内，则它必定垂直于这个法向量，因此点积为零。

3. Using the Determinant Form | 使用行列式形式

In coordinates, the scalar triple product is conveniently computed as a 3×3 determinant whose rows are the components of the three vectors.

在坐标形式下，标量三重积可以方便地表示为一个 3×3 行列式，其行（或列）为三个向量的分量。

If a = (a₁, a₂, a₃), b = (b₁, b₂, b₃), c = (c₁, c₂, c₃), then:

设 a = (a₁, a₂, a₃)，b = (b₁, b₂, b₃)，c = (c₁, c₂, c₃)，则：

a · (b × c) = det( a₁ a₂ a₃ ; b₁ b₂ b₃ ; c₁ c₂ c₃ )

Expanding the determinant gives the same scalar value. If this value is zero, the vectors are linearly dependent, which is exactly equivalent to coplanarity.

展开行列式即可得到相同的标量值。如果该值为零，则向量线性相关，这正好等价于共面性。

4. Geometric Interpretation | 几何解释

The absolute value of the scalar triple product equals the volume of the parallelepiped formed by the three vectors. If the vectors are coplanar, the parallelepiped collapses to a flat shape, so its volume is zero.

标量三重积的绝对值等于由三个向量构成的平行六面体的体积。如果三个向量共面，平行六面体就会坍缩成一个平面图形，因此体积为零。

This mental picture helps you remember the condition: coplanarity means zero volume, and zero volume means zero scalar triple product.

这个直观图像有助于记忆该条件：共面意味着体积为零，体积为零意味着标量三重积为零。

5. Coplanarity of Points | 点的共面性

Sometimes the question asks whether four points A, B, C, D are coplanar. This is equivalent to checking whether the three vectors AB, AC and AD are coplanar.

有时题目会问四个点 A、B、C、D 是否共面。这等价于检验三个向量 AB、AC 和 AD 是否共面。

So you first form the position vectors from one point to the other three, then test their scalar triple product.

因此，你首先从一个点出发构造到另外三个点的向量，然后检验它们的标量三重积。

(B − A) · [(C − A) × (D − A)] = 0 ⇔ A, B, C, D are coplanar

If the result is zero, the four points lie on the same plane; otherwise they do not.

如果结果为零，则四个点位于同一平面；否则它们不共面。

6. Worked Example | 例题详解

Let a = (1, 2, 3), b = (2, −1, 0), c = (4, 3, 6). Determine whether these vectors are coplanar.

设 a = (1, 2, 3)，b = (2, −1, 0)，c = (4, 3, 6)。判断这三个向量是否共面。

First compute b × c:

首先计算 b × c：

b × c = ( (−1)(6) − (0)(3), (0)(4) − (2)(6), (2)(3) − (−1)(4) ) = ( −6, −12, 10 )

Then compute the dot product with a:

然后计算与 a 的点积：

a · (b × c) = 1(−6) + 2(−12) + 3(10) = −6 − 24 + 30 = 0

Since the scalar triple product is zero, the vectors are coplanar.

由于标量三重积为零，这三个向量共面。

7. Using the Determinant for the Same Example | 用行列式重解上例

Alternatively, set up the determinant directly:

或者直接构造行列式：

det( 1 2 3 ; 2 −1 0 ; 4 3 6 )

Expanding along the first row:

按第一行展开：

1 × det( −1 0 ; 3 6 ) − 2 × det( 2 0 ; 4 6 ) + 3 × det( 2 −1 ; 4 3 )

Compute each minor: ( −1×6 − 0×3 ) = −6; ( 2×6 − 0×4 ) = 12; ( 2×3 − (−1)×4 ) = 10.

计算各余子式：( −1×6 − 0×3 ) = −6；( 2×6 − 0×4 ) = 12；( 2×3 − (−1)×4 ) = 10。

So the determinant is: 1(−6) − 2(12) + 3(10) = −6 − 24 + 30 = 0. Hence coplanar.

因此行列式为：1(−6) − 2(12) + 3(10) = −6 − 24 + 30 = 0。故共面。

8. Relation to Linear Dependence | 与线性相关的关系

Three vectors in 3D are coplanar if and only if they are linearly dependent. This means one of them can be written as a linear combination of the other two, provided the other two are not parallel.

三维空间中的三个向量共面当且仅当它们线性相关。这意味着其中一个向量可以写成另外两个向量的线性组合（前提是另外两个向量不平行）。

For example, if c = αa + βb, then c lies in the plane spanned by a and b, so the three vectors are coplanar.

例如，如果 c = αa + βb，则 c 位于由 a 和 b 张成的平面内，因此三个向量共面。

The scalar triple product being zero is exactly the determinant condition for linear dependence of three vectors.

标量三重积为零正是三个向量线性相关的行列式条件。

9. Special Cases and Warnings | 特殊情况与注意事项

If any of the vectors is the zero vector, the scalar triple product is automatically zero, because the volume is zero. In this case, the vectors are considered coplanar.

如果其中任何向量为零向量，标量三重积自动为零，因为体积为零。在这种情况下，这些向量被视为共面。

If two of the vectors are parallel, the cross product of those two is zero, so the scalar triple product is also zero. This is still a valid coplanarity result.

如果其中两个向量平行，那么这两个向量的叉积为零，因此标量三重积也为零。这仍然是有效的共面性结论。

Be careful not to confuse coplanarity of vectors with coplanarity of lines or points. Always identify what exactly the question gives you.

注意不要将向量的共面性与直线或点的共面性混淆。务必明确题目给出的具体对象是什么。

10. Geometric Method Using Plane Equation | 利用平面方程的几何方法

Another way to test coplanarity of points is to find the equation of the plane through three of the points, then check whether the fourth point satisfies that equation.

检验点共面的另一种方法是：先求出经过其中三个点的平面方程，然后检查第四个点是否满足该方程。

If the plane through A, B, C has equation r · n = d, then point D is coplanar with A, B, C if and only if D · n = d.

如果经过 A、B、C 的平面方程为 r · n = d，则点 D 与 A、B、C 共面当且仅当 D · n = d。

Here n can be found as AB × AC, and d = A · n. This method is often more intuitive in exam questions about points.

其中 n 可取 AB × AC，d = A · n。在处理点的题型时，这种方法往往更直观。

11. Common AQA Exam Questions | 常见 AQA 考试题型

AQA A-Level questions often ask you to:

AQA A-Level 题目通常要求你：

Determine whether three given vectors are coplanar using the scalar triple product.
测试三个给定向量是否共面（使用标量三重积）。
Show that four points lie in the same plane.
证明四个点位于同一平面。
Find a value of a parameter such that three vectors become coplanar.
求某参数的值，使得三个向量共面。

For parameter problems, set the determinant equal to zero and solve for the unknown. This gives a single equation in one variable (or more if several parameters appear).

对于含参数的问题，令行列式等于零并解出未知数。这通常会得到一个一元方程（若有多个参数则会得到更多方程）。

12. Summary | 总结

To test coplanarity of vectors a, b, c, compute the scalar triple product a · (b × c). If it is zero, the vectors are coplanar; otherwise they are not.

要检验向量 a、b、c 的共面性，计算标量三重积 a · (b × c)。若为零，则向量共面；否则不共面。

For points A, B, C, D, test whether (B − A) · [(C − A) × (D − A)] = 0. This is the same idea applied to position vectors.

对于点 A、B、C、D，检验 (B − A) · [(C − A) × (D − A)] = 0。这是同一思想在位置向量上的应用。

Mastering this determinant and its geometric meaning will help you solve both pure and applied questions involving planes and 3D geometry.

掌握这一行列式及其几何意义，将帮助你在纯数学和应用题中轻松解决涉及平面和三维几何的问题。

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Coplanarity of Vectors | 向量的共面性