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  2. Rotation matrix - Wikipedia

    en.wikipedia.org/wiki/Rotation_matrix

    A 180° rotation (middle) followed by a positive 90° rotation (left) is equivalent to a single negative 90° (positive 270°) rotation (right). Each of these figures depicts the result of a rotation relative to an upright starting position (bottom left) and includes the matrix representation of the permutation applied by the rotation (center ...

  3. Rotation of axes in two dimensions - Wikipedia

    en.wikipedia.org/wiki/Rotation_of_axes_in_two...

    In mathematics, a rotation of axes in two dimensions is a mapping from an xy-Cartesian coordinate system to an x′y′-Cartesian coordinate system in which the origin is kept fixed and the x′ and y′ axes are obtained by rotating the x and y axes counterclockwise through an angle .

  4. Rotation (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Rotation_(mathematics)

    The rotation is acting to rotate an object counterclockwise through an angle θ about the origin; see below for details. Composition of rotations sums their angles modulo 1 turn, which implies that all two-dimensional rotations about the same point commute. Rotations about different points, in general, do not commute.

  5. 3D rotation group - Wikipedia

    en.wikipedia.org/wiki/3D_rotation_group

    Every non-trivial rotation is determined by its axis of rotation (a line through the origin) and its angle of rotation. Rotations are not commutative (for example, rotating R 90° in the x-y plane followed by S 90° in the y-z plane is not the same as S followed by R), making the 3D rotation group a nonabelian group.

  6. Quaternions and spatial rotation - Wikipedia

    en.wikipedia.org/wiki/Quaternions_and_spatial...

    3D visualization of a sphere and a rotation about an Euler axis (^) by an angle of In 3-dimensional space, according to Euler's rotation theorem, any rotation or sequence of rotations of a rigid body or coordinate system about a fixed point is equivalent to a single rotation by a given angle about a fixed axis (called the Euler axis) that runs through the fixed point. [6]

  7. Fantasy football Start 'Em, Sit 'Em: 40 players to start or ...

    www.aol.com/fantasy-football-start-em-sit...

    Week 12 marks the first "Byemageddon” of the NFL season in fantasy football. A season-high six teams have their bye this week: the New York Jets, Atlanta Falcons, Buffalo Bills, Cincinnati ...

  8. Elon Musk efficiency panel seeks 'high IQ' staff, plans ...

    www.aol.com/news/elon-musk-efficiency-panel...

    Elon Musk's government efficiency panel wants "high IQ" employees and plans weekly livestreams, according to X posts about President-elect Donald Trump's initiative to streamline the U.S. bureaucracy.

  9. Rodrigues' rotation formula - Wikipedia

    en.wikipedia.org/wiki/Rodrigues'_rotation_formula

    Let k be a unit vector defining a rotation axis, and let v be any vector to rotate about k by angle θ (right hand rule, anticlockwise in the figure), producing the rotated vector . Using the dot and cross products, the vector v can be decomposed into components parallel and perpendicular to the axis k,