Consider the cube with vertices 111 Give standard matrix of

Consider the cube with vertices (+-1,+-1,+-1) . Give standard matrix of the given symmetries of the cube. a) 90 degree rotation about the z axis b) 180 degree rotation about the line joining (-1,0,1) and (1,0,-1) c) 120 degree rotation about the line joining (-1,-1,-1) and (1,1,1)

Solution

a) Rotaiton about 90 degree about z-axis is given by

T =

cos 90 sin 90 0 0
-sin 90 cos 90 0 0
0 0 1 0
0 0 0 1

b) Run this matlab code.

d=1.732;
cx= 0.7071;
cy=0;
cz=0.354;
Ralpa =[1 0 0 0 ;0 cz/d cy/d 0; 0 -cy/d cz/d 0;0 0 0 1];
Rbeta = [d 0 cx 0;0 1 0 0;-cx 0 d 0;0 0 0 1];
Tr=[1 0 0 0; 0 1 0 0; 0 0 1 0;1 0 -1 1];
R=[cos(180) sin(180) 0 0 ; -sin(180) cos(180) 0 0 ;0 0 1 0;0 0 0 1];
a=inv(Rbeta);
b=inv(Ralpa);
c=inv(Tr);
T= Tr*Ralpa*Rbeta*R*a*b*c

(rotation 180 degree about line (-1,0,1) and (1,0,-1)

T =

-0.3701 -6.7890 2.7367 0
0.0810 -0.5985 -0.1619 0
0.1143 0.5665 0.7716 0
-1.4844 -7.3555 2.9651 1.0000

C)120 degree rotation about line joining (-1,-1,-1) and (1,1,1 ) is given by

Run this matlab code:

d=1.732;
cx= 0.5774;
cy=0.5774;
cz=0.5774;
Ralpa =[1 0 0 0 ;0 cz/d cy/d 0; 0 -cy/d cz/d 0;0 0 0 1];
Rbeta = [d 0 cx 0;0 1 0 0;-cx 0 d 0;0 0 0 1];
Tr=[1 0 0 0; 0 1 0 0; 0 0 1 0;1 0 -1 1];
R=[cos(120) sin(180) 0 0 ; -sin(180) cos(180) 0 0 ;0 0 1 0;0 0 0 1];
a=inv(Rbeta);
b=inv(Ralpa);
c=inv(Tr);
T= Tr*Ralpa*Rbeta*R*a*b*c

T =

0.8328 -1.9975 2.1648 0
0.1574 0.3534 0.4893 0
-0.1202 1.0906 0.0296 0
-0.0470 -3.0882 3.1352 1.0000