1) Consider any two body-fixed coordinate systems. F_(B1) and F_(B21) where F_(B2) is related to F_(g1) by a positive rotation about the y_(a1) exis through an angle Theta (a) Determine the nine angles between the axes of F_(B1) and those of F_(B2) Use this information to write down T_(B2,81) using the direction cosine matrix (b) For a 321 Euler angle sequence from F_(B1) to F_(B2) determine the three angles Theta _(1),Theta _(1) and Theta _(2) Use this information to write down T_(B2,81) using the Fuller angle representation of the direction cosine matrix (c) Determine the elgenaxis and angle that relates F_(B2) to F_(BI) Use this information to write down T_(B2,81) using the Euler parameter representation of the direction cosine matrix

(a) To determine the nine angles between the axes of and those of , we need to consider the rotation described. Since is related to by a positive rotation about the axis through an angle , the angles between the axes are:- Between and : - Between and : - Between and : - Between and : - Between and : - Between and : - Between and : - Between and : - Between and : Using this information, we can write down using the direction cosine matrix as: (b) For a 321 Euler angle sequence from to , the three angles , , and can be determined as follows:- : This is the angle between and , which is .- : This is the angle between and , which is .- : This is the angle between and , which is .Using this information, we can write down using the Fuller angle representation of the direction cosine matrix as: (c) To determine the eigenaxis and angle that relates to , we need to find the eigenvector and eigenvalue of the transformation matrix . The eigenaxis is the eigenvector corresponding to the eigenvalue of that is equal to 1. The angle is the angle between the eigenaxis and the axis.The eigenvalues of are and 1. The corresponding eigenvectors are and , respectively.The eigenaxis is , and the angle between this eigenaxis and the axis is .Using this information, we can write down using the Euler parameter representation of the direction cosine matrix as: