Rotation Matrices

Part 1.  Trigonometric Background

Let (x,y) denote the terminal point of the vector x in the plane.  Suppose we rotate this vector through an angle theta, and let ((Subscript[x, r]), (Subscript[y, r])) denote the new terminal point after the rotation.  Our fiirst objective is  to calculate the new coordinates in terms of the old ones.

We can write x = b*cos(psi ) and y=b*sin(psi), where b is the length of the vector x.  The new coordinates are:

x[r]  =  b*Cos[θ + ψ]y[r]  =  b*Sin[θ + ψ]

TrigExpand[x[r]]TrigExpand[y[r]]

Enter your matrix A here by completing the following definition:

A  =  { {?, ?}, {?, ?} };  A // MatrixForm

Part 2.  Two-Dimensional Rotation Matrices

Enter (Subscript[e, 1]), the image (Subscript[v, 1]) when (Subscript[e, 1]) is rotated 60 degrees, and the rotation matrix B.  Check that B (Subscript[e, 1]) = (Subscript[v, 1]).

e1  =  {1, 0}\(v1\  = \ {\(?,?}\)

        B\  = \ {{\(?,?},{?,?}};\ B\ //\ MatrixForm\)\)

Adapt the preceding definitions to check the image of (Subscript[e, 2]).

Part 3.  Three-Dimensional Rotation Matrices

Define rotation matrices around each of the axes.

\(P\  = 
      \ {\ {Cos[θ], \ \(-Sin[θ]\), \ 0}, 
        \ {Sin[θ], \ Cos[θ], \ 0}, \ 

        		\ \ \ \ \ \ \ \ {0, 0, 1}\ }; \ \ P\  // \ MatrixForm\)\(Q\  = 
      \ {\ {1, 0, 0}, \ {0, \ Cos[θ], \ \(-Sin[θ]\)}, \ 

        		\ \ \ \ \ \ \ \ {0, \ Sin[θ], \ Cos[θ]}\ }; \ \ 
    Q\  // \ MatrixForm\)\(R\  = 
      \ {\ {Cos[θ], \ 0, \ Sin[θ]}, \ {0, 1, 0}, 

        		\ \ \ \ \ \ \ \ {\(-Sin[θ]\), \ 0, \ Cos[θ]}\ }; 
    \ \ R\  // \ MatrixForm\)

Plot an object in 3-space.

\(Plot3D[4\  - \ 2*x1\  - \ 2*x2, \ {x1, \ 0, \ 2}, \ {x2, \ 0, \ 2}, \ 

      	\ \ \ \ \ \ \ \ \ \ \ \ \ \ AxesLabel\  -> \ {x1, \ x2, \ x3}\ , \ 

      	\ \ \ \ \ ViewPoint\  -> \ {\(-2\), \ 0, 0}\ ]\)

Part 4.  Determinants and Inverses of Rotation Matrices

Redefine the general rotation matrix in the plane.

A  = 
       { {Cos[θ],  -Sin[θ]}, 
         {Sin[θ],  Cos[θ]} };   A // MatrixForm

Part 5.  Summary