IB Mathematics: Rotational Transformations in Three-Dimensional Coordinates | IB数学：三维坐标轴旋转变换

📚 IB Mathematics: Rotational Transformations in Three-Dimensional Coordinates | IB数学：三维坐标轴旋转变换

In three-dimensional coordinate geometry, a rotation transformation moves a point or vector around a fixed axis while preserving its distance from the origin. Unlike translations, rotations do not change the shape, size, or orientation of an object; they only change its direction in space. In IB Mathematics, particularly in the Analysis and Approaches Higher Level and Applications and Interpretation courses, understanding rotation matrices is essential for working with vectors, transformations, and geometric modelling.

在三维坐标几何中，旋转变换使点或向量绕固定轴转动，同时保持其到原点的距离不变。与平移不同，旋转不会改变物体的形状、大小或方向，只会改变它在空间中的朝向。在IB数学中，尤其是在数学分析与方法高级课程以及数学应用与解释课程中，理解旋转矩阵对于处理向量、变换和几何建模至关重要。

1. Right-Hand Rule and Sign Conventions | 右手定则与符号约定

Before writing rotation matrices, we must agree on the positive direction of rotation. In a standard right-handed three-dimensional coordinate system, align the thumb of your right hand with the positive direction of an axis. The curl of your fingers then shows the direction of a positive rotation about that axis. For example, a positive rotation about the z-axis moves the positive x-axis toward the positive y-axis.

在写出旋转矩阵之前，我们必须约定旋转的正方向。在标准右手三维坐标系中，将右手拇指指向某个轴的正方向，其余手指弯曲的方向就是绕该轴正向旋转的方向。例如，绕z轴正向旋转时，x轴正方向会转向y轴正方向。

About the x-axis: positive rotation turns y toward z.

绕x轴：正向旋转使y转向z。
About the y-axis: positive rotation turns z toward x.

绕y轴：正向旋转使z转向x。
About the z-axis: positive rotation turns x toward y.

绕z轴：正向旋转使x转向y。

2. The General Form of a 3D Rotation Matrix | 三维旋转矩阵的一般形式

A rotation in three dimensions is represented by a 3 × 3 matrix \(R\) that is orthogonal and has determinant 1. When we multiply a column vector \(\mathbf{v}\) by \(R\), we obtain the rotated vector \(\mathbf{v}’ = R\mathbf{v}\). The matrix form is concise and allows us to combine multiple rotations by matrix multiplication.

三维旋转可以用一个3×3矩阵 \(R\) 表示，该矩阵是正交矩阵且行列式为1。当我们用 \(R\) 乘以列向量 \(\mathbf{v}\) 时，就得到旋转后的向量 \(\mathbf{v}’ = R\mathbf{v}\)。矩阵形式非常简洁，并且可以通过矩阵乘法将多次旋转组合起来。

For a rotation by angle θ about the x-axis, the transformation is:

绕x轴旋转角度θ的变换为：

x’	=	x
y’	=	y cos θ − z sin θ
z’	=	y sin θ + z cos θ

Equivalently, in matrix notation:

等价的矩阵记法为：

1	0	0
0	cos θ	−sin θ
0	sin θ	cos θ

3. Rotation About the x-Axis | 绕x轴旋转

The rotation matrix about the x-axis, denoted \(R_x(\theta)\), acts on a vector by leaving the x-coordinate unchanged. Only the y- and z-coordinates are mixed together. This is useful when rotating an object around a horizontal axis, such as tilting a plane forward or backward.

绕x轴的旋转矩阵记为 \(R_x(\theta)\)，它作用在向量上时保持x坐标不变，只混合y和z坐标。这在绕水平轴旋转物体时非常有用，例如使平面向前或向后倾斜。

For \(R_x(\theta)\):

对于 \(R_x(\theta)\)：

R_x(θ) =	1	0	0
	0	cos θ	−sin θ
	0	sin θ	cos θ

Example: Rotate the point (2, 0, 1) by 90° about the x-axis. Since cos 90° = 0 and sin 90° = 1, we get x’ = 2, y’ = 0 − 1 = −1, z’ = 0 + 0 = 0. The image is (2, −1, 0).

示例：将点(2, 0, 1)绕x轴旋转90°。由于cos 90° = 0，sin 90° = 1，得到x’ = 2，y’ = 0 − 1 = −1，z’ = 0 + 0 = 0。旋转后的点是(2, −1, 0)。

4. Rotation About the y-Axis | 绕y轴旋转

For rotation about the y-axis, the y-coordinate remains fixed, while the x- and z-coordinates transform. The matrix contains the sine terms in opposite positions compared with the x-axis rotation. This is often used to “yaw” an object left or right in a horizontal plane.

绕y轴旋转时，y坐标保持不变，x和z坐标发生变换。与绕x轴旋转相比，矩阵中正弦项的位置相反。这种旋转常用于使物体在水平面内左右转动（偏航）。

For \(R_y(\theta)\):

对于 \(R_y(\theta)\)：

R_y(θ) =	cos θ	0	sin θ
	0	1	0
	−sin θ	0	cos θ

Applying \(R_y(\theta)\) to a vector (x, y, z):

将 \(R_y(\theta)\) 应用于向量(x, y, z)：

x’ = x cos θ + z sin θ, y’ = y, z’ = −x sin θ + z cos θ

5. Rotation About the z-Axis | 绕z轴旋转

Rotation about the z-axis is the simplest and most intuitive three-dimensional rotation, because it looks exactly like a two-dimensional rotation in the xy-plane. The z-coordinate is unaffected.

绕z轴旋转是最简单、最直观的三维旋转，因为它看起来就像xy平面内的二维旋转。z坐标不受影响。

For \(R_z(\theta)\):

对于 \(R_z(\theta)\)：

R_z(θ) =	cos θ	−sin θ	0
	sin θ	cos θ	0
	0	0	1

The corresponding coordinate equations are:

对应的坐标方程为：

x’ = x cos θ − y sin θ, y’ = x sin θ + y cos θ, z’ = z

6. Properties of Rotation Matrices | 旋转矩阵的性质

All rotation matrices share a set of important algebraic properties. Recognising these properties helps you verify whether a given matrix actually represents a rotation, and it enables you to find inverse rotations quickly.

所有旋转矩阵都具有一系列重要的代数性质。识别这些性质有助于你验证给定矩阵是否真正表示旋转，并帮助你快速求出逆旋转。

Orthogonality: \(R^T R = I\), where \(I\) is the 3 × 3 identity matrix.

正交性： \(R^T R = I\)，其中 \(I\) 是三阶单位矩阵。
Determinant: det(R) = 1. A reflection matrix has determinant −1, so it is not a rotation.

行列式： det(R) = 1。反射矩阵的行列式为−1，因此不是旋转矩阵。
Length preservation: \(\|R\mathbf{v}\| = \|\mathbf{v}\|\) for every vector \(\mathbf{v}\).

长度保持： 对任意向量 \(\mathbf{v}\)，有 \(\|R\mathbf{v}\| = \|\mathbf{v}\|\)。
Inverse equals transpose: \(R^{-1} = R^T\), because \(R^T R = I\).

逆等于转置： \(R^{-1} = R^T\)，因为 \(R^T R = I\)。

7. Combining Rotations: Order Matters | 复合旋转：顺序至关重要

If you rotate a point first about the x-axis by angle α and then about the y-axis by angle β, the combined transformation is not the same as rotating first about the y-axis and then about the x-axis. Matrices are generally not commutative: \(R_x(α) R_y(β) \neq R_y(β) R_x(α)\).

如果先将点绕x轴旋转角度α，再绕y轴旋转角度β，复合变换与先绕y轴再绕x轴的结果不同。矩阵通常不可交换： \(R_x(α) R_y(β) \neq R_y(β) R_x(α)\)。

Suppose the first rotation is \(R_1\) and the second rotation is \(R_2\). For a column vector \(\mathbf{v}\), the final vector is:

假设第一次旋转为 \(R_1\)，第二次旋转为 \(R_2\)。对于列向量 \(\mathbf{v}\)，最终向量为：

\(\mathbf{v}’ = R_2 R_1 \mathbf{v}\)

The matrix closest to the vector is applied first. In general, the overall rotation matrix is \(R_{\text{total}} = R_n \cdots R_2 R_1\), where the leftmost matrix is the last rotation applied. Always write the order explicitly in your working.

最靠近向量的矩阵最先作用。一般来说，总旋转矩阵为 \(R_{\text{total}} = R_n \cdots R_2 R_1\)，其中最左边的矩阵是最后应用的旋转。在解题过程中务必明确写出作用顺序。

8. Inverse Rotation and the Transpose | 逆旋转与转置

The inverse of a rotation by angle θ is a rotation by angle −θ. Because rotation matrices are orthogonal, the inverse matrix is simply the transpose. Therefore \(R^{-1}(θ) = R^T(θ) = R(−θ)\). This gives a very fast way to “undo” a rotation.

角度θ旋转的逆旋转就是角度−θ的旋转。由于旋转矩阵是正交矩阵，其逆矩阵就是转置矩阵。因此 \(R^{-1}(θ) = R^T(θ) = R(−θ)\)。这为“撤销”旋转提供了非常快捷的方法。

For example, for the z-axis:

例如，绕z轴旋转：

R_z(θ)^{-1} =	cos θ	sin θ	0
	−sin θ	cos θ	0
	0	0	1

This is identical to \(R_z(−θ)\), since cos(−θ) = cos θ and sin(−θ) = −sin θ.

这正好等于 \(R_z(−θ)\)，因为cos(−θ) = cos θ，sin(−θ) = −sin θ。

9. Rotating a Point versus Rotating the Axes | 旋转点与旋转坐标轴

In many physics and geometry problems, you must distinguish between an active rotation and a passive rotation. An active rotation moves the vector while the coordinate axes stay fixed. A passive rotation keeps the vector fixed but rotates the coordinate axes. These two descriptions are mathematically equivalent but with opposite signs.

在许多物理和几何问题中，必须区分主动旋转和被动旋转。主动旋转是保持坐标轴不动而移动向量；被动旋转是保持向量不动而旋转坐标轴。这两种描述在数学上是等价的，但旋转角度符号相反。

If the coordinate axes are rotated by angle θ about an axis, the coordinates of a fixed point in the new system are obtained by applying the rotation matrix for angle −θ to the original coordinates. In IB questions, always read carefully whether the axes are being rotated or the point itself is being rotated.

如果坐标轴绕某轴旋转角度θ，那么在新坐标系中，固定点的坐标等于对原坐标施加角度−θ的旋转矩阵。在IB题目中，务必仔细阅读题目是旋转坐标轴还是旋转点本身。

10. Euler Angles: A Brief Introduction | 欧拉角：简要介绍

Euler angles describe any three-dimensional rotation as a sequence of three rotations about different axes. A common convention, called ZXZ, uses a rotation about the z-axis, then about the new x-axis, then about the new z-axis again. Although Euler angles are beyond the core IB syllabus, they connect naturally to rotation matrices and appear in many university-level applications.

欧拉角将任意三维旋转描述为绕不同轴的三次旋转序列。一种常用的约定称为ZXZ顺序：先绕z轴旋转，再绕新的x轴旋转，最后再绕新的z轴旋转。虽然欧拉角超出了IB核心大纲，但它与旋转矩阵自然关联，并出现在许多大学层面的应用中。

The overall matrix is the product of three basic rotation matrices, but the middle rotation is about the new axis after the first rotation. This is why the matrix product is not simply \(R_z R_x R_z\) in the original coordinate system; the order and the reference frame must be stated clearly.

总矩阵是三个基本旋转矩阵的乘积，但中间那次旋转是绕第一次旋转之后的新轴进行的。因此总矩阵并不是简单地在原坐标系中写成 \(R_z R_x R_z\)；必须明确说明顺序和参考系。

11. Worked Example: A 90° Rotation About the z-Axis | 例题：绕z轴旋转90°

Question: Find the image of the point A(1, 2, 3) after a rotation of 90° about the positive z-axis.

问题：求点A(1, 2, 3)绕z轴正方向旋转90°后的坐标。

Solution: For θ = 90°, cos θ = 0 and sin θ = 1. We apply the z-axis rotation equations:

解答：对于θ = 90°，cos θ = 0，sin θ = 1。应用绕z轴旋转方程：

x’ = x cos θ − y sin θ, y’ = x sin θ + y cos θ, z’ = z

Substitute x = 1, y = 2, z = 3:

代入x = 1，y = 2，z = 3：

x’ = 1 × 0 − 2 × 1 = −2

y’ = 1 × 1 + 2 × 0 = 1

z’ = 3

Therefore the image is A'(−2, 1, 3). You can check that the distance from the origin is preserved: original length = √(1² + 2² + 3²) = √14, new length = √((−2)² + 1² + 3²) = √14.

因此旋转后的点为A'(−2, 1, 3)。可以验证到原点的距离保持不变：原长度 = √(1² + 2² + 3²) = √14，新长度 = √((−2)² + 1² + 3²) = √14。

12. Common Mistakes and IB Exam Tips | 常见错误与IB考试建议

Many students lose marks on rotation questions because of small sign errors or incorrect ordering. Below are the most common pitfalls and how to avoid them.

很多学生在旋转题目中因为符号错误或顺序错误而失分。以下是最常见的陷阱以及如何避免。

Wrong sine sign: In \(R_x(\theta)\) and \(R_z(\theta)\), the negative sine appears in the top row; in \(R_y(\theta)\), it appears in the bottom row. Memorise each matrix separately rather than guessing.

正弦符号错误：在 \(R_x(\theta)\) 和 \(R_z(\theta)\) 中，负正弦项出现在第一行；在 \(R_y(\theta)\) 中，负正弦项出现在第三行。请分别记忆每个矩阵，不要随意猜测。
Using degrees instead of radians: IB exam questions may expect radians. If your calculator is in degree mode, your answer will be wrong when the angle is given in radians. Always check the mode.

用度数代替弧度：IB考试题目可能要求使用弧度。如果计算器处于度数模式，而角度以弧度给出，结果就会出错。务必检查计算器模式。
Wrong multiplication order: When combining rotations, the first rotation is multiplied on the right. Writing \(R_1 R_2\) when you mean \(R_2 R_1\) changes the answer.

乘法顺序错误：组合旋转时，第一次旋转写在右边。将 \(R_1 R_2\) 写成 \(R_2 R_1\) 会改变结果。
Confusing active and passive rotations: Rotating the coordinate axes by +θ is equivalent to rotating the vector by −θ. Read the problem statement carefully.

混淆主动与被动旋转：坐标轴旋转+θ等价于向量旋转−θ。请仔细阅读题目表述。
Forgetting to verify the result: After computing a rotated vector, check that the length is unchanged. This quick check catches many numerical errors.

忘记验证结果：计算旋转后的向量后，检查长度是否不变。这一快速检查能发现许多数值错误。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

IB Mathematics: Rotational Transformations in Three-Dimensional Coordinates | IB数学：三维坐标轴旋转变换