The rotationmatrix [R], associated with a positive (in a right-hand sense) rotation α ; about the z-axis is:

cosα sinα 0

-sinα cosα 0

0 0 1

? Derive the rotation Matrix [Q] relating (x, y, z) to a system (x', y', z') which is described by three consecutive Euler rotations about z, then y then x.

? Show that [Q] is orthogonal.

? The transformation rule for the strain tensor under a rotation is is [ε'] = [Q] [ε ] [Q]T . Why is not possible to find a matrix [A] such that [ε'] = [Q] [ε ] [Q]T Write down three quantities that you know to be invariant under the rotation.

? Write a Matlab program or construct an Excel spreadsheet to calculate the strain tensor under an arbitrary rotation in terms of given strains in a global (x,y,z) system, and to calculate the strain in-variants. Calculate the rotated strain tensor for Euler rotations of (45, 30, 30) degrees of the global strain tensor
[-0.1 0.05 0.0, 0.05 0.3 0.0, 0.0 0.0 -0.1]

? Calculate the strain in-variants in both systems, showing that they are same.
? Describe the deformed characteristics and sketch the deformation of a cube of material subjected to this strain.

Given the 2x2 matrices A, B, and C (active transformation matrices) in the x, y plane do the following:
(A, B and C are 2x2 matrices given below)
1. Show thematrix is orthogonal
2. Find the determinant to indicate if a rotation or reflection matrix
3. Find therotation angle or find the line of reflection.
A= 1/(2)

1. show that for a 2x2 unitary matrix its determinant is a complex number of unit modulus
2. Verify that the Pi/2 rotationmatrix is orthogonal.
3. Verify that the matrices:
A= 1/sqrt(2) * ( 1 i )
( i 1)
and:
B = 1/2 * ( 1+i 1-i )
( 1-i 1+i )
are unitary. Verify

We use matrices, eigenvalues and eigenvectors to solve the following:
Let f be therotation in the 3-dimensional space about the x-axis through the angle pi/2 and g be therotation about the y-axis through the same angle. Describe therotation f g. Let h be the reflection in the plane orthogonal
to the vector 2i - j + 3k. D

Question 4:
Give an example of a 2 by 2 rotationmatrix that rotates the point
( 4 , 5 )
into the square with corners
( -3 , -7 )
( -1 , -7 )
( -1 , -5 )
( -3 , -5 )
You will probably need to experiment to find a suitable rotation. You can c

Assume the operator R(theta) transforms a vector v (in two-dimensional real space) by swinging it around counterclockwise through an angle theta. How can I show that the operator R(theta) has thematrix representation
cos(theta) -sin(theta)
sin(theta) cos(theta)
That is supposed to be a matrix.

Let T1 : R2 → R2 and T2 : R2 → R2 be thematrix transformations induced by the matrices M1 and M2 respectively, where
M1= -1 0 and M2= 0 -1
0 1 -1 0
Find the general formula for the transformations T1 and T2, and determine the geometric meaning of T1 and T2.
Find thematrix o

A diver can change her rate of rotation in the aair by "tucking" her head in and bending her knees. Let's assume that when she is stretched out straight she is rotating at 1 revolution per second. Now she goes into the "tuck and bend", effectively shortening the length of her body by half. What will her rate of rotation be now?

The serial manipulator shown in figure 1 (in the attachment) is set in a configuration such that the pose of the effector with respect to a world reference frame, Rworld, is defined by the composed homogeneous transformation matrix, Qgripper/world, that follows (defined with respect to evolving reference frames):
{Please refe

Please see the attachment for a diagram of the robot in question which is a 3-degrees of freedom manipulator with prismatic joints, as well as the following questions:
a) Assign the X and Z axes and determine the Denavit-Hartenberg parameters of this robot.
b) Determine the A matrices of this robot.
c) What is the position