2. Three-dimensional rotation of vectors can be descried by 3 × 3 matrices. A rotationaround the x-axis by angle 0 is given by the matrix O_{x}(\theta)=\left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & \cos \theta & \sin \theta \\ 0 & -\sin \theta & \cos \theta \end{array}\right) A rotation around the y-axis by angle ø is given by the matrix O_{y}(\phi)=\left(\begin{array}{ccc} \cos \phi & 0 & \sin \phi \\ 0 & 1 & 0 \\ -\sin \phi & 0 & \cos \phi \end{array}\right) and a rotation around the z-axis by angle x is given by the matrix O_{z}(\chi)=\left(\begin{array}{ccc} \cos \chi & \sin \chi & 0 \\ -\sin \chi & \cos \chi & 0 \\ 0 & 0 & 1 \end{array}\right) a) The vector V is given by \mathbf{V}=\left(\begin{array}{l} 0 \\ 0 \\ 1 \end{array}\right) Show by explicit calculations that O_{z}(\chi=\pi) O_{y}(\phi=\pi / 2) \mathbf{V}=O_{y}(\phi=-\pi / 2) \mathbf{V} Explain with words the geometric interpretation of this equation. Evaluate the composite rotation O2(x1)Oz(x2) and express it as a rotation around the z axis with respect to a single angle. By using the iteration procedure, determine0:(x), where n is a positive integer, in terms of a rotation with respect to a single angle.[6] Find the general angles x and phi of two successive rotations of the form O2(x)O,(phi)acting on vector V such that the resulting vector corresponds to a rotation Ox(0)of V around the x-axis for a given angle phi. Draw the corresponding diagram that depicts the rotations of V with respect to the angles theta, phi and x.[8]

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2. Three-dimensional rotation of vectors can be descried by 3 × 3 matrices. A rotationaround the x-axis by angle 0 is given by the matrix O_{x}(\theta)=\left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & \cos \theta & \sin \theta \\ 0 & -\sin \theta & \cos \theta \end{array}\right) A rotation around the y-axis by angle ø is giv