Refer to the figure above showing two right$-$handed Cartesian coordinate frames; one is frame

$0$ and the other is frame $3.$ A vector is expressed in frame $0,[$X$0$ Y$0$ Z$0],$ and it is required to

transform the vector to frame $3.$ That is$,$ we want the vector components in frame $3,[$X$3$ Y$3$

Z$3].$ Develop transformation matrices to perform the transformation from frame $0$ to frame $3.$

Find the transformation matrices for three rotation sequences:

a$)3-1-2$ rotation sequence

b$)1-2-3$ rotation sequence

c$)2-1-3$ rotation sequence