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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 .
To rotate about any horizontal axis, simply subtract from that axis from each formula. If h is the value of a horizontal axis, then the volume equals ((()) (())). For example, to rotate the region between y = −2x + x 2 and y = x along the axis y = 4, one would integrate as follows:
For a change of basis, the formula of the preceding section applies, with the same change-of-basis matrix on both sides of the formula. That is, if M is the square matrix of an endomorphism of V over an "old" basis, and P is a change-of-basis matrix, then the matrix of the endomorphism on the "new" basis is .
It is also used for calculating stresses in many planes by reducing them to vertical and horizontal components. These are called principal planes in which principal stresses are calculated; Mohr's circle can also be used to find the principal planes and the principal stresses in a graphical representation, and is one of the easiest ways to do ...
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.
The above procedure now is reversed to find the form of the function F(x) using its (assumed) known log–log plot. To find the function F, pick some fixed point (x 0, F 0), where F 0 is shorthand for F(x 0), somewhere on the straight line in the above graph, and further some other arbitrary point (x 1, F 1) on the same graph.
By rotating the cube by 45° on the x-axis, the point (1, 1, 1) will therefore become (1, 0, √ 2) as depicted in the diagram. The second rotation aims to bring the same point on the positive z-axis and so needs to perform a rotation of value equal to the arctangent of 1 ⁄ √ 2 which is approximately 35.264°.
For an xyz-Cartesian coordinate system in three dimensions, suppose that a second Cartesian coordinate system is introduced, with axes x', y' and z' so located that the x' axis is parallel to the x axis and h units from it, the y' axis is parallel to the y axis and k units from it, and the z' axis is parallel to the z axis and l units from it.