• HW for 5/21

Posted by Allen Olsen on 5/21/2019

Objective: Derive the equations of conic sections (parabola and ellipse); continue to do solve problems in Linear Algebra.

HW: Problem Set 3 in the yellow book.

• HW for 5/14

Posted by Allen Olsen on 5/14/2019

Your homework is based on what we did today in class.  It is in this PDF.

• HW for 5/9

Posted by Allen Olsen on 5/8/2019

Objective: Derive Rodrigo's formulas and use them to find the axis and angle of a combined rotation.

Classwork / HW: This revised worksheet.

Also, here is the 2/7 worksheet and here is the 2/11 worksheet.

• HW for 5/7

Posted by Allen Olsen on 5/7/2019

Objective: Use Euler's Theorem and Rodrigo's Formulas to find the axis and angle of the resultant rotation when one roation follows another.

Euler's Theorem says that a rotation through a unit axis m by β radians followed by a rotation through a unit axis l (that's a lower-case L) by α radians is equivalent to a rotation through a third unit axis n by γ radians.

To find n and γ from l and m and α and β, place the unit vectors at the center of a unit sphere.  On the surface of the sphere, let A be the point determined by l and let B be the point determined by m.  Connect A and B with an arc of a great circle.  Rotate the arc AB by α/2 radians counterclockwise around A to form another arc, and rotate AB by β/2 radians clockwise around B to form a third arc, intersecting the second arc at C.  This forms a spherical triangle ABC.  Call the angle at C, π - γ/2. Then Euler's Theorem says that the counterclockwise rotation by β at B, followed by a ccw rotation by α at A, is equivalent to a ccw rotation by γ at C.

Rodrigo used this information to construct formulas for figuring out the axis n and the angle γ of the combined rotation.  First, he defined the following quantities: λ_A = cos (1/2 α), and Λ_A = sin (1/2 α) l, with similar expressions for λ_B and Λ_B.

Then his formulas are:

λ_C = λ_A · λ_B - (Λ_A · Λ_B), and

Λ_C = λ_A · Λ_B + λ_B · Λ_A + (Λ_A × Λ_B).

HW:

Use the two formulas above, together with the definitions.

1. Show that, if A is the rotation R(π/3 i) and B is the rotation R(π/6 k), that λ_A = √(3) / 2 and λ_B = (√(6)+√(2)) / 4.

2. Show that Λ_A = (1/2) i and Λ_B = [(√(6)-√(2)) / 4 ] k.

3. Show that λ_C = (3√(2) + √(6)) / 8 and thus that cos (γ) = (3√(3) - 2) / 8, as we found before.

4. Show that Λ_C = [(√(6)+√(2)) / 8] times < 1, -2 + √(3), 2√(3) - 3>, which is proportional to the axis vector we found in class.

5. You may wish to review the Law of Cosines (for sides and angles), and the Law of Sines, for spherical triangles before the next class so that I don't catch you by surprise!

• HW for 5/6

Posted by Allen Olsen on 5/6/2019

Objective: Use matrices to perform three-dimensional rotations.

HW: Today we did a problem where we were doing first a rotation around the z-axis by +30 degrees, and then a rotation around the x-axis by +60 degrees.

We determined that the combined rotation was for 66.452 degrees, and were working on determining a vector that was pointing in the direction of the axis.

1. Verify that the vector <1, -(2 - √(3)), 2√(3)-3 > points in the direction of the axis of rotation for the combined rotation.  Do this by multiplying it by the matrix, and getting the vector itself back again.

2. Now go back to what we were doing in class.  See if you can solve the system of equations to get this vector.  Remember that you can only solve for two of x, y, z in terms of one of them.

• HW for 4/30

Posted by Allen Olsen on 4/30/2019

Objective: Apply the properties of orthogonal matrices to three-dimensional rotations.

HW: In the handout, read sections 1.3 and 1.4 and do problems 1.13 - 1.15.  Please come with your questions tomorrow.

• HW for 4/29

Posted by Allen Olsen on 4/29/2019

Objective: Use properties of orthogonal matrices to do 2- and 3-dimensional rotations and reflections.

HW: By this point you should be through section 1.1 of the handout and you should understand how to do exercises 1.1 to 1.5.  Tonight, read section 1.2 on Similarity Transforms, and do exercises 1.6 through 1.12.

Important: if you are struggling with some of the exercises, make a worthy attempt.  When we go through them in class, go back to the problems you were having trouble with, and redo them -- rework the answers to reflect your greater understanding.  These problems may need several passes.  You have the time to do a careful job.

It will be useful for you to make a fair copy of your answers when you are satisfied with them.  This will be a valuable reference for you later.

• HW for 4/22

Posted by Allen Olsen on 4/22/2019

No homework tonight -- we didn't get far enough into the material about orthogonal matrices.  Please review your notes today so that you can ask questions in class tomorrow.

The assignment tomorrow will be longish, so please allow time to complete it before Thursday.

That's all for now.

• Solutions for HW for 4/9

Posted by Allen Olsen on 4/10/2019

Here are the solutions to the last problem set.  We will go over these in class Thursday.

• HW for 4/9

Posted by Allen Olsen on 4/9/2019

HW: This is one of the more unusual homework assignments I have given.

Watch out!