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The vector projection (also known as the vector component or vector resolution) of a vector a on (or onto) a nonzero vector b is the orthogonal projection of a onto a straight line parallel to b. The projection of a onto b is often written as proj b a {\displaystyle \operatorname {proj} _{\mathbf {b} }\mathbf {a} } or a ∥ b .
The Euler parameters can be viewed as the coefficients of a quaternion; the scalar parameter a is the real part, the vector parameters b, c, d are the imaginary parts. Thus we have the quaternion Thus we have the quaternion
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 R v . A more direct method, however, is to simply calculate the trace : the sum of the diagonal elements of the rotation matrix.
(See 2:1 mapping of SU(2) to SO(3)) If both rotations are a half-turn (=), both q and −q will have a real coordinate equal to zero. Otherwise, one will have a positive real part, representing a rotation by an angle less than , and the other will have a negative real part, representing a rotation by an angle greater than .
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.
Illustration of the vector formulation. The equation of a line can be given in vector form: = + Here a is the position of a point on the line, and n is a unit vector in the direction of the line. Then as scalar t varies, x gives the locus of the line.
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 such a presentation, the notions of length and angle are defined by means of the dot product. The length of a vector is defined as the square root of the dot product of the vector by itself, and the cosine of the (non oriented) angle between two vectors of length one is defined as their dot product. So the equivalence of the two definitions ...