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The vector projection of a on b is a vector a 1 which is either null or parallel to b. More exactly: a 1 = 0 if θ = 90°, a 1 and b have the same direction if 0° ≤ θ < 90°, a 1 and b have opposite directions if 90° < θ ≤ 180°.
To find the angle of a rotation, once the axis of the rotation is known, select a vector v perpendicular to the axis. Then the angle of the rotation is the angle between v and Rv. A more direct method, however, is to simply calculate the trace: the sum of the diagonal elements of the rotation
The vector cross product, used to define the axis–angle representation, does confer an orientation ("handedness") to space: in a three-dimensional vector space, the three vectors in the equation a × b = c will always form a right-handed set (or a left-handed set, depending on how the cross product is defined), thus fixing an orientation in ...
The angle θ and axis unit vector e define a rotation, concisely represented by the rotation vector θe.. In mathematics, the axis–angle representation parameterizes a rotation in a three-dimensional Euclidean space by two quantities: a unit vector e indicating the direction of an axis of rotation, and an angle of rotation θ describing the magnitude and sense (e.g., clockwise) of the ...
In the theory of three-dimensional rotation, Rodrigues' rotation formula, named after Olinde Rodrigues, is an efficient algorithm for rotating a vector in space, given an axis and angle of rotation. By extension, this can be used to transform all three basis vectors to compute a rotation matrix in SO(3) , the group of all rotation matrices ...
Angular distance or angular separation is the measure of the angle between the orientation of two straight lines, rays, or vectors in three-dimensional space, or the central angle subtended by the radii through two points on a sphere.
Phasor notation (also known as angle notation) is a mathematical notation used in electronics engineering and electrical engineering.A vector whose polar coordinates are magnitude and angle is written . [13] can represent either the vector (, ) or the complex number + =, according to Euler's formula with =, both of which have magnitudes of 1.
In mathematics and mechanics, the Euler–Rodrigues formula describes the rotation of a vector in three dimensions. It is based on Rodrigues' rotation formula , but uses a different parametrization. The rotation is described by four Euler parameters due to Leonhard Euler .