Question

8. Given the following sets of equations $x' = \frac{\sqrt{2}}{2}(x+y), y' = \frac{\sqrt{2}}{2}(-x+y), and z' = z$ and $x'' = \frac{\sqrt{2}}{2}(-x'+y'), y'' = -\frac{\sqrt{2}}{2}(x'+y'), and z'' = z'$ (a) write each set of equation using matrices (b) prove that the coefficient matrices corresponding to the two sets of equations can correspond to rotation matrices by showing that the matrices are orthogonal matrices. (c) solve for $x''$ and $y''$ in terms of x and y by multiplying the coefficient matrices for the two sets of equation. (d) the two sets of equations represent rotations of axes about the z-axis. By comparing the matrices for the two set of equations with the rotation matrix derived in class, find the rotation angles for each set. 3 Length L Period T 00:01.69 Figure 2: Simple pendulum of length L. (e) using the matrix you found in (c) find the angle of rotation from the x - y to directly to x' - y' and show that it is equal to the sum of the rotations angles you determined in part (d).

          8. Given the following sets of equations

$x' = \frac{\sqrt{2}}{2}(x+y), y' = \frac{\sqrt{2}}{2}(-x+y), and z' = z$
and
$x'' = \frac{\sqrt{2}}{2}(-x'+y'), y'' = -\frac{\sqrt{2}}{2}(x'+y'), and z'' = z'$

(a) write each set of equation using matrices
(b) prove that the coefficient matrices corresponding to the two sets of equations can correspond to rotation matrices
by showing that the matrices are orthogonal matrices.
(c) solve for $x''$ and $y''$ in terms of x and y by multiplying the coefficient matrices for the two sets of equation.
(d) the two sets of equations represent rotations of axes about the z-axis. By comparing the matrices for the two
set of equations with the rotation matrix derived in class, find the rotation angles for each set.
3
Length L
Period T
00:01.69
Figure 2: Simple pendulum of length L.
(e) using the matrix you found in (c) find the angle of rotation from the x - y to directly to x' - y' and show that
it is equal to the sum of the rotations angles you determined in part (d).

8. Given the following sets of equations

x' = (√(2))/(2)(x+y), y' = (√(2))/(2)(-x+y), and z' = z
and
x” = (√(2))/(2)(-x'+y'), y” = -(√(2))/(2)(x'+y'), and z” = z'

(a) write each set of equation using matrices
(b) prove that the coefficient matrices corresponding to the two sets of equations can correspond to rotation matrices
by showing that the matrices are orthogonal matrices.
(c) solve for x” and y” in terms of x and y by multiplying the coefficient matrices for the two sets of equation.
(d) the two sets of equations represent rotations of axes about the z-axis. By comparing the matrices for the two
set of equations with the rotation matrix derived in class, find the rotation angles for each set.
3
Length L
Period T
00:01.69
Figure 2: Simple pendulum of length L.
(e) using the matrix you found in (c) find the angle of rotation from the x - y to directly to x' - y' and show that
it is equal to the sum of the rotations angles you determined in part (d).

Added by Brandy E.

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