(a) Prove that, for all θ, φ, ψ, Q=( cosφcosψ-cosθsinφsinψ sinφcosψ+cosθcosφsinψ sinθsin

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(a) Prove that, for all θ, φ, ψ, Q=( ...

(a) Prove that, for all $\theta, \varphi, \psi$, $$ Q=\left(\begin{array}{ccc} \cos \varphi \cos \psi-\cos \theta \sin \varphi \sin \psi & \sin \varphi \cos \psi+\cos \theta \cos \varphi \sin \psi & \sin \theta \sin \psi \\ -\cos \varphi \sin \psi-\cos \theta \sin \varphi \cos \psi & -\sin \varphi \sin \psi+\cos \theta \cos \varphi \cos \psi & \sin \theta \cos \psi \\ \sin \theta \sin \varphi & -\sin \theta \cos \varphi & \cos \theta \end{array}\right) $$ is a proper orthogonal matrix. (b) Write down a formula for $Q^{-1}$. Remark. It can be shown that every proper orthogonal matrix can be parameterized in this manner; $\theta, \varphi, \psi$ are known as the Euler angles, and play an important role in applications in mechanics and geometry, $[31 ;$ p. 147].

(a) Prove that, for all $\theta, \varphi, \psi$,
$$
Q=\left(\begin{array}{ccc}
\cos \varphi \cos \psi-\cos \theta \sin \varphi \sin \psi & \sin \varphi \cos \psi+\cos \theta \cos \varphi \sin \psi & \sin \theta \sin \psi \\
-\cos \varphi \sin \psi-\cos \theta \sin \varphi \cos \psi & -\sin \varphi \sin \psi+\cos \theta \cos \varphi \cos \psi & \sin \theta \cos \psi \\
\sin \theta \sin \varphi & -\sin \theta \cos \varphi & \cos \theta
\end{array}\right)
$$
is a proper orthogonal matrix. (b) Write down a formula for $Q^{-1}$.
Remark. It can be shown that every proper orthogonal matrix can be parameterized in this manner; $\theta, \varphi, \psi$ are known as the Euler angles, and play an important role in applications in mechanics and geometry, $[31 ;$ p. 147].

Key Concepts

Proper Orthogonal Matrix

A proper orthogonal matrix is a square matrix whose columns (and rows) form an orthonormal set, and whose determinant is +1. This means that the matrix preserves both lengths and angles when applied to vectors, which is especially important in applications involving rotations. Because of these properties, proper orthogonal matrices are widely used to represent rotations in Euclidean space without reflection.

Euler Angles

Euler angles are a set of three parameters that describe the orientation of a rigid body or coordinate system in three-dimensional space. They break down a rotation into three successive rotations about specified axes. This parameterization allows for a continuous and unique representation of any possible rotation, making them fundamental in mechanics and computer graphics for describing spatial orientations.

Matrix Orthogonality and Inversion

Matrix orthogonality refers to the property that a matrix Q satisfies Q^T Q = I, where I is the identity matrix. This property ensures that the matrix preserves the dot product and, consequently, the lengths and angles of vectors. An important consequence of orthogonality is that the inverse of an orthogonal matrix is simply its transpose (Q^{-1} = Q^T), which greatly simplifies computations related to rotations and their inversions in various applications within linear algebra and geometry.

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