Example 12
Matrices Applied to Two Dimensional
Translations and Rotations

Copyright © 1993 by Bill Titus, Carleton College, 
Department of Physics and Astronomy, Northfield, MN  55057-4025
September 6, 1993

Topics and Skills

Problem

Solution - Part a

Step 1 - Using the fact that matrices within MMA are lists of lists, the following code defines symbolic matrices for  r,  ro,  and  rot[c].  Look it over and then activate the cell.

Clear[x, y, xo, yo, c, r, ro, rot]
r  = {x, y}
ro = {-xo, -yo}
rot[c_] = {{Cos[c], Sin[c]}, {-Sin[c], Cos[c]}}

Comment 1

Step 2 - Display the matrices in "standard form" using the MMA function   MatrixForm[].

Comment 2

Solution - Part b

Step 1 - Show that a translation followed by a rotation is not the same as a rotation followed by a translation.  Note that within MMA, matrix multiplication is indicated by a period.  Be sure to simplify your results so that you can easily see that the operations don't commute.

Solution - Part c

Step 1 - Find the inverse of  rot[c]  and call it  rotInv[c].  What MMA command do you think you're going to use?

Comment 1

Step 2 - Redefine  rotInv[c]  using  Simplify[]  to make it "pretty" and then display the result in matrix form.  Postfix notation may be useful here.

Comment 2

Step 3 - Verify that  rotInv[c]  is indeed the inverse of  rot[c]  by showing that their product is the identity matrix.

Comment 3

Solution - Part d

Step 1 - Show that two consecutive rotations by angles  c1  and  c2  are the same as a single rotation by the angle  c1 + c2.

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