answer:Solution: # Part (a): Composition of Two Symmetries 1. **Introduction of Symmetries:** Let us consider two axial symmetries of space with axes ( s_1 ) and ( s_2 ), which intersect at point ( O ) and form an angle ( varphi = s_1 widehat{O} s_2 ). 2. **Decomposition of Symmetries into Planar Symmetries:** Each of these axial symmetries can be represented as a composition of two symmetries with respect to planes. Consider the pairs of planes ( alpha_1 perp alpha_2 ) and ( beta_1 perp beta_2 ), which represent the symmetries corresponding to axes ( s_1 ) and ( s_2 ), respectively. 3. **Choice of Intersecting Planes:** We choose the planes ( alpha_2 ) and ( beta_1 ) to be the plane of intersection of the axes ( s_1 O s_2 ). 4. **Identity Transformation Step:** The composition of the symmetries with respect to coinciding planes ( alpha_2 ) and ( beta_1 ) is the identity transformation. This gives us the clue on how symmetries with respect to the remaining planes should be handled. 5. **Resultant Transformation as Rotation:** The composition of the remaining symmetries produces a rotation about an axis ( l ), which is perpendicular to the plane determined by ( s_1 O s_2 ) and passes through ( O ). The angle of this rotation is twice the angle ( varphi ), making it ( 2varphi ). 6. **Construction of Rotation Axis and Angle:** The rotation axis ( l ) is constructed as described, and the angle of rotation can be determined to be ( 2varphi ). # Part (b): Representation of Rotations 1. **Equivalence of Rotation and Composition of Symmetries:** As previously shown, any rotation can be represented as a composition of two axial symmetries. 2. **Intersection of Axes:** The axes of these symmetries intersect at a point on the rotation axis and form an angle that is half of the rotation angle. 3. **Conclusion:** Specifically, if we have a rotation about some axis by an angle ( theta ), this rotation can be realized as the composition of two symmetries about intersecting axes forming an angle ( theta/2 ). Thus, we have demonstrated that the composition of two axis symmetries with intersecting axes results in a rotation and that any rotation can be decomposed into such a composition. blacksquare

answer:Let P = left( 1, theta, frac{pi}{3} right), then its rectangular coordinates (x, y, z) are derived from the spherical to rectangular coordinate conversion, which yields: [x = rho sin phi cos theta,] [y = rho sin phi sin theta,] [z = rho cos phi.] Plugging in rho = 1, phi = frac{pi}{3}: [x = sin frac{pi}{3} cos theta = frac{sqrt{3}}{2} cos theta,] [y = sin frac{pi}{3} sin theta = frac{sqrt{3}}{2} sin theta,] [z = cos frac{pi}{3} = frac{1}{2}.] Then, the radius of the circle formed by the projection onto the xy-plane is: [sqrt{x^2 + y^2} = sqrt{left(frac{sqrt{3}}{2} cos thetaright)^2 + left(frac{sqrt{3}}{2} sin thetaright)^2} = sqrt{frac{3}{4} (cos^2 theta + sin^2 theta)} = frac{sqrt{3}}{2}.] Thus, the radius of the circle is boxed{frac{sqrt{3}}{2}}.

answer:To solve this problem, we can try to find a pattern by calculating the first few terms of the sequence. Given (a_1 = frac{1}{2}), we can calculate (a_2) as follows: [a_2 = frac{a_1}{a_1 + 1} = frac{frac{1}{2}}{frac{1}{2} + 1} = frac{frac{1}{2}}{frac{3}{2}} = frac{1}{3}] Continuing this process, we calculate (a_3): [a_3 = frac{a_2}{a_2 + 1} = frac{frac{1}{3}}{frac{1}{3} + 1} = frac{frac{1}{3}}{frac{4}{3}} = frac{1}{4}] From these calculations, we can observe a pattern that (a_n = frac{1}{n+1}). Therefore, for (a_{2018}), we have: [a_{2018} = frac{1}{2018 + 1} = frac{1}{2019}] Thus, the correct answer is boxed{text{D}}.

answer:1. **Projection Calculation**: Project begin{pmatrix} x y end{pmatrix} onto begin{pmatrix} 3 4 end{pmatrix}: [ operatorname{proj}_{begin{pmatrix} 3 4 end{pmatrix}} begin{pmatrix} x y end{pmatrix} = frac{begin{pmatrix} x y end{pmatrix} cdot begin{pmatrix} 3 4 end{pmatrix}}{begin{pmatrix} 3 4 end{pmatrix} cdot begin{pmatrix} 3 4 end{pmatrix}} begin{pmatrix} 3 4 end{pmatrix} ] [ = frac{3x + 4y}{25} begin{pmatrix} 3 4 end{pmatrix} = begin{pmatrix} frac{9x + 12y}{25} frac{12x + 16y}{25} end{pmatrix}. ] 2. **Reflection Calculation**: Reflecting across begin{pmatrix} 3 4 end{pmatrix} after projecting (not a standard transformation but valid as a compound): [ begin{pmatrix} x' y' end{pmatrix} = 2 times begin{pmatrix} frac{9x + 12y}{25} frac{12x + 16y}{25} end{pmatrix} - begin{pmatrix} x y end{pmatrix} ] [ = begin{pmatrix} frac{18x + 24y}{25} - x frac{24x + 32y}{25} - y end{pmatrix} = begin{pmatrix} frac{18x + 24y - 25x}{25} frac{24x + 32y - 25y}{25} end{pmatrix} = begin{pmatrix} frac{-7x + 24y}{25} frac{24x + 7y}{25} end{pmatrix}. ] Hence, the matrix for the reflected projection transformation is: [ boxed{begin{pmatrix} -7/25 & 24/25 24/25 & 7/25 end{pmatrix}}. ]