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In mathematics and physics, the right-hand rule is a convention and a mnemonic, utilized to define the orientation of axes in three-dimensional space and to determine the direction of the cross product of two vectors, as well as to establish the direction of the force on a current-carrying conductor in a magnetic field.
Example problem based on Shepard & Metzlar's "Mental Rotation Task": are these two three-dimensional shapes identical when rotated? Mental rotation is the ability to rotate mental representations of two-dimensional and three-dimensional objects as it is related to the visual representation of such rotation within the human mind. [1]
Rotation formalisms are focused on proper (orientation-preserving) motions of the Euclidean space with one fixed point, that a rotation refers to.Although physical motions with a fixed point are an important case (such as ones described in the center-of-mass frame, or motions of a joint), this approach creates a knowledge about all motions.
The conjugacy definition would also allow a mirror image of the structure, but this is not needed, the structure itself is achiral. For example, if a symmetry group contains a 3-fold axis of rotation, it contains rotations in two opposite directions. (The structure is chiral for 11 pairs of space groups with a screw axis.)
[2] [3] [a] In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations, such as translation, reflection, rotation, or scaling. Although these two meanings of the word can sometimes be told apart, they are intricately related, and hence are discussed together in ...
For example, in 2-space n = 2, a rotation by angle θ has eigenvalues λ = e iθ and λ = e −iθ, so there is no axis of rotation except when θ = 0, the case of the null rotation. In 3-space n = 3, the axis of a non-null proper rotation is always a unique line, and a rotation around this axis by angle θ has eigenvalues λ = 1, e iθ, e −iθ.
The X axis is now at angle γ with respect to the x axis. The XYZ system rotates again, but this time about the x axis by β. The Z axis is now at angle β with respect to the z axis. The XYZ system rotates a third time, about the z axis again, by angle α. In sum, the three elemental rotations occur about z, x and z.
Some examples of true 6DOF games, which allow independent control of all three movement axes and all three rotational axes, include Elite Dangerous, Shattered Horizon, the Descent franchise, the Everspace franchise, Retrovirus, Miner Wars, Space Engineers, Forsaken and Overload (from the same creators of Descent).