Question

Show that if \(H\) is a proper orthogonal matrix such that \(H_{33}=1\), then there is a unique angle \(\alpha \in[0,2 \pi)\) such that $$ H=\left(\begin{array}{ccc} \cos \alpha & \sin \alpha & 0 \\ -\sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1 \end{array}\right) $$ Show that if \(H_{33}=-1\), then there is a unique angle \(\alpha \in[0,2 \pi)\) such that $$ H=\left(\begin{array}{ccc} -\cos \alpha & -\sin \alpha & 0 \\ -\sin \alpha & \cos \alpha & 0 \\ 0 & 0 & -1 \end{array}\right) $$ Sketch a diagram of two orthonormal triads with this transition matrix, showing the angle \(\alpha\).

Short answer

Answer: When H33 = 1, the matrix H is given by: $$ H = \begin{pmatrix} \cos\alpha & \sin\alpha & 0\\ -\sin\alpha & \cos\alpha & 0\\ 0 & 0 & 1 \end{pmatrix} $$ When H33 = -1, the matrix H is given by: $$ H = \begin{pmatrix} -\cos\alpha & -\sin\alpha & 0\\ -\sin\alpha & \cos\alpha & 0\\ 0 & 0 & -1 \end{pmatrix} $$ In both cases, α represents the angle between the initial and final positions, resulting from rotation (and reflection for H33 = -1) in the xy-plane.

Step-by-step solution

Consider Orthogonal Matrix Properties

Let's start by recalling some properties of orthogonal matrices. An orthogonal matrix H has the property that \(H^TH=HH^T=I\), where I is the identity matrix.

Check the condition for the cases H33=1 and H33=-1

Now, we need to check the condition where H is a proper orthogonal matrix with \(H_{33}=1\). We can use the property that diagonal elements of an orthogonal matrix must satisfy the following condition: \(H_{11}^2 + H_{12}^2 + H_{13}^2 = 1\). For the case \(H_{33} = 1\), we can ignore the third row and third column, focusing on the upper-left 2x2 sub-matrix. For the case where \(H_{33} = -1\), we need to analyze the other two rows and columns in a similar manner, while taking note that the third element will have a different sign.

Find α for the case H33=1

For the upper-left 2x2 sub-matrix, we can write it as: $$ \begin{pmatrix} H_{11} & H_{12}\\ H_{21} & H_{22} \end{pmatrix} $$ Using the property that \(H^TH = I\), we get the following equations: $$ H_{11}^2 + H_{12}^2 = 1\\ H_{21}^2 + H_{22}^2 = 1\\ H_{11}H_{21} + H_{12}H_{22} = 0 $$ Based on these equations, we can represent H11 as \(\cos\alpha\), H12 as \(\sin\alpha\), H21 as \(-\sin\alpha\) and H22 as \(\cos\alpha\). Therefore, H can be written as: $$ H = \begin{pmatrix} \cos\alpha & \sin\alpha & 0\\ -\sin\alpha & \cos\alpha & 0\\ 0 & 0 & 1 \end{pmatrix} $$

Find α for the case H33=-1

Similarly, when \(H_{33} = -1\), we can see that the upper-left 2x2 sub-matrix is still orthogonal, leading to the same equations as before. The main difference is that the third element on the diagonal will be -1 this time. Considering our orthogonal conditions, we can represent the matrix as: $$ H = \begin{pmatrix} -\cos\alpha & -\sin\alpha & 0\\ -\sin\alpha & \cos\alpha & 0\\ 0 & 0 & -1 \end{pmatrix} $$

Sketch the orthonormal triads

For the case \(H_{33} = 1\), the matrix representation shows a rotation of angle α in the xy-plane, while keeping the z-axis fixed. For the case \(H_{33} = -1\), the matrix representation shows a rotation of angle α in the xy-plane along with a reflection in the xy-plane. The sketch should consist of two orthonormal triads where the angle α represents the angle between the initial and final positions. The first orthonormal triad can be represented by a rotation in the xy-plane, while the second one can be represented by a rotation along with a reflection in the xy-plane.

Key concepts

Analytical Dynamics and Orthogonal Matrices

Analytical dynamics is a branch of classical mechanics that deals with the study of motion of objects using analytical methods. It prominently features mathematical tools such as matrices to describe rotations and transformations in space. One fascinating aspect within analytical dynamics is the concept of orthogonal matrices.

An orthogonal matrix, denoted as H, is a square matrix with real entries that preserves the length of vectors during transformations. In simpler terms, it represents a rotation or a reflection. The defining property of an orthogonal matrix is that the product of the matrix and its transpose results in the identity matrix \( I \), expressed mathematically as \( H^T H = HH^T = I \). When applied to the dynamics of a system, these matrices can describe the motion without changing the object's shape or size—crucial in understanding the kinematics and kinetics of particles and rigid bodies.

Matrix Representation in Dynamics

Matrix representation is a powerful tool in physics and engineering that facilitates the description of linear transformations, such as changes in coordinates due to rotation or reflection. In the context of orthogonal matrices, we use a matrix representation to convey the behavior of a system where directions may change, but the initial magnitudes of vectors are preserved.

In the example provided, the matrix H is used to represent a rotation for the case where \( H_{33} = 1 \) and a rotation followed by a reflection for the case where \( H_{33} = -1 \). Both cases conserve the lengths of vectors, which is a clear indication of orthogonal transformation. The angle of rotation \( \alpha \) within the matrix is crucial for understanding the behavior of the system and how it evolves from one state to another. By expressing transformations using matrix notation, students can analyze and visualize complex movements in a way that is both systematic and conceptually clear.

Understanding Orthonormal Triads in Physical Transformations

Orthonormal triads are sets of three vectors in space that are both orthogonal (at right angles to each other) and normalized (each vector has a unit length). They serve as a fundamental concept in physics and engineering for defining coordinate systems and describing spatial orientations.

In the exercise, we encountered two kinds of matrices that affect orthonormal triads: One represents a simple rotation where the z-axis remains untouched \( H_{33} = 1 \) and the other encompasses both a rotation and a reflection \( H_{33} = -1 \). These actions can be visualized if one imagines a set of axes moving: for rotation, the x- and y-axes swirl around the z-axis by an angle \( \alpha \), but in the case of rotation and reflection, an additional flip across the xy-plane occurs, effectively inverting the direction of the z-axis.

Understanding these concepts is crucial in fields such as robotics, aerospace, and mechanical engineering, where spatial orientation and transformations directly impact the design and control of complex systems.