Application of Vector Products to Areas | 向量积在面积中的应用

📚 Application of Vector Products to Areas | 向量积在面积中的应用

The vector product, or cross product, is a powerful tool in 3D geometry. One of its most useful applications is calculating the areas of parallelograms and triangles when only vectors or coordinates are known. This article explains how to use the magnitude of the cross product to find areas, with step-by-step worked examples.

向量积（叉积）是三维几何中的强大工具。它最有用的应用之一，是在只知道向量或坐标时计算平行四边形和三角形的面积。本文将讲解如何利用叉积的模求面积，并配有逐步解题示例。

---

1. The Vector Product Defined | 向量积的定义

The vector product of two vectors a and b, written as a × b, is a vector perpendicular to both a and b. Its magnitude is given by

|a × b| = |a||b| sin θ

where θ is the angle between the two vectors.

两个向量 a 和 b 的向量积写作 a × b，是一个同时垂直于 a 和 b 的向量。其模为

|a × b| = |a||b| sin θ

其中 θ 是两向量之间的夹角。

The direction of a × b is determined by the right-hand rule. For parallel or anti-parallel vectors, θ is 0° or 180°, so sin θ = 0 and the cross product is the zero vector.

a × b 的方向由右手定则确定。当两向量平行或反平行时，θ 为 0° 或 180°，所以 sin θ = 0，叉积为零向量。

---

2. Area of a Parallelogram | 平行四边形的面积

In a parallelogram, let two adjacent sides be represented by vectors a and b. The perpendicular height is |b| sin θ, so the area is base × height = |a| × |b| sin θ = |a × b|.

在平行四边形中，设两条相邻边分别对应向量 a 和 b。垂直高为 |b| sin θ，因此面积为 底 × 高 = |a| × |b| sin θ = |a × b|。

Therefore, to find the area of a parallelogram, simply compute the magnitude of the cross product of its two side vectors.

因此，求平行四边形的面积，只需要计算其两条边向量的叉积的模。

---

3. Area of a Triangle | 三角形的面积

A triangle can be viewed as half of a parallelogram, so its area is half the magnitude of the cross product of two side vectors.

三角形可以看作平行四边形的一半，因此其面积等于两条边向量的叉积模的一半。

Area = ½ |a × b|

This formula works for any triangle in 2D or 3D space, as long as a and b are two non-zero side vectors sharing a common vertex.

只要 a 和 b 是共顶点的非零边向量，该公式适用于二维或三维空间中的任何三角形。

---

4. Using Position Vectors for a Triangle | 利用位置向量求三角形面积

Given a triangle whose vertices have position vectors a, b and c, take two side vectors as b − a and c − a.

已知三角形三个顶点的位置向量分别为 a、b 和 c，取两条边向量为 b − a 和 c − a。

Then the area of the triangle is ½ |(b − a) × (c − a)|. This approach avoids the need to know the angle between the sides explicitly.

则三角形面积为 ½ |(b − a) × (c − a)|。这种方法无需显式知道两边夹角。

---

5. Worked Example — Triangle Area in 3D | 例 1 —— 三维三角形面积

Find the area of the triangle with vertices A(1,2,3), B(2,0,1) and C(-1,1,4).

求顶点为 A(1,2,3)、B(2,0,1)、C(-1,1,4) 的三角形面积。

Let a, b, c be the position vectors. Then AB = b − a = (1, −2, −2) and AC = c − a = (−2, −1, 1).

设 a、b、c 为位置向量。则 AB = b − a = (1, −2, −2)，AC = c − a = (−2, −1, 1)。

Their cross product is

AB × AC = (−4, 3, −5)

因为 AB = (1, −2, −2) 且 AC = (−2, −1, 1)，所以叉积为

AB × AC = (−4, 3, −5)

The magnitude is √[(−4)² + 3² + (−5)²] = √50 = 5√2, so the area of the triangle is ½ × 5√2 = (5√2) / 2 square units.

其模为 √[(−4)² + 3² + (−5)²] = √50 = 5√2，因此三角形面积为 ½ × 5√2 = (5√2) / 2 平方单位。

---

6. Worked Example — Parallelogram Area | 例 2 —— 平行四边形面积

Find the area of the parallelogram with adjacent sides p = 2i + j − 3k and q = i − 2j + k.

求以 p = 2i + j − 3k 和 q = i − 2j + k 为相邻边的平行四边形面积。

First compute p × q:

p × q = (−5, −5, −5)

先计算 p × q：

p × q = (−5, −5, −5)

The magnitude is √[(−5)² + (−5)² + (−5)²] = √75 = 5√3, which is the area of the parallelogram.

其模为 √[(−5)² + (−5)² + (−5)²] = √75 = 5√3，这就是平行四边形的面积。

---

7. Component Formula for the Magnitude | 叉积模的分量公式

If a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃), then the square of the magnitude of the cross product is expressed in component form as

|a × b|² = (a₂b₃ − a₃b₂)² + (a₃b₁ − a₁b₃)² + (a₁b₂ − a₂b₁)²

若 a = (a₁, a₂, a₃)，b = (b₁, b₂, b₃)，则叉积模的平方可写为分量形式

|a × b|² = (a₂b₃ − a₃b₂)² + (a₃b₁ − a₁b₃)² + (a₁b₂ − a₂b₁)²

This form is particularly useful when the angle between vectors is not easy to compute, and it gives exact algebraic results.

当两向量夹角不易计算时，这种形式特别有用，并能得到精确的代数结果。

---

8. Geometric Interpretation and Sign | 几何意义与符号

The vector a × b is perpendicular to the plane containing a and b. Its length equals the area of the parallelogram, and its direction indicates the orientation of that plane.

向量 a × b 垂直于由 a 和 b 所在的平面。其长度等于平行四边形面积，其方向表明该平面的朝向。

Because area is a positive scalar, we always use the magnitude. Swapping the order of a and b changes the

Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）