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 $F_{B1}$ and those of $F_{B2}$, we need to consider the rotation described. Since $F_{B2}$ is related to $F_{B1}$ by a positive rotation about the $y_{a1}$ axis through an angle $\Theta$, the angles between the axes are:<br /><br />- Between $x_{a1}$ and $x_{a2}$: $\Theta$<br />- Between $x_{a1}$ and $y_{a2}$: $0$<br />- Between $x_{a1}$ and $z_{a2}$: $0$<br />- Between $y_{a1}$ and $x_{a2}$: $-\Theta$<br />- Between $y_{a1}$ and $y_{a2}$: $0$<br />- Between $y_{a1}$ and $z_{a2}$: $0$<br />- Between $z_{a1}$ and $x_{a2}$: $0$<br />- Between $z_{a1}$ and $y_{a2}$: $-\Theta$<br />- Between $z_{a1}$ and $z_{a2}$: $0$<br /><br />Using this information, we can write down $T_{B2,81}$ using the direction cosine matrix as:<br /><br />$T_{B2,81} = \begin{bmatrix}<br />\cos(\Theta) & 0 & 0 \\<br />0 & 1 & 0 \\<br />0 & 0 & \cos(\Theta)<br />\end{bmatrix}$<br /><br />(b) For a 321 Euler angle sequence from $F_{B1}$ to $F_{B2}$, the three angles $\Theta_1$, $\Theta_2$, and $\Theta_3$ can be determined as follows:<br /><br />- $\Theta_1$: This is the angle between $x_{a1}$ and $x_{a2}$, which is $\Theta$.<br />- $\Theta_2$: This is the angle between $y_{a1}$ and $y_{a2}$, which is $0$.<br />- $\Theta_3$: This is the angle between $z_{a1}$ and $z_{a2}$, which is $\Theta$.<br /><br />Using this information, we can write down $T_{B2,81}$ using the Fuller angle representation of the direction cosine matrix as:<br /><br />$T_{B2,81} = \begin{bmatrix}<br />\cos(\Theta) & 0 & 0 \\<br />0 & 1 & 0 \\<br />0 & 0 & \cos(\Theta)<br />\end{bmatrix}$<br /><br />(c) To determine the eigenaxis and angle that relates $F_{B2}$ to $F_{BI}$, we need to find the eigenvector and eigenvalue of the transformation matrix $T_{B2,81}$. The eigenaxis is the eigenvector corresponding to the eigenvalue of $T_{B2,81}$ that is equal to 1. The angle is the angle between the eigenaxis and the $x_{a1}$ axis.<br /><br />The eigenvalues of $T_{B2,81}$ are $\cos(\Theta)$ and 1. The corresponding eigenvectors are $\begin{bmatrix}<br />\cos(\Theta) & 0 & -\sin(\Theta)<br />\end{bmatrix}$ and $\begin{bmatrix}<br />0 & 1 & 0<br />\end{bmatrix}$, respectively.<br /><br />The eigenaxis is $\begin{bmatrix}<br />\cos(\Theta) & 0 & -\sin(\Theta)<br />\end{bmatrix}$, and the angle between this eigenaxis and the $x_{a1}$ axis is $\Theta$.<br /><br />Using this information, we can write down $T_{B2,81}$ using the Euler parameter representation of the direction cosine matrix as:<br /><br />$T_{B2,81} = \begin{bmatrix}<br />\cos(\Theta) & 0 & -\sin(\Theta) \\<br />0 & 1 & 0 \\<br />\sin(\Theta) & 0 & \cos(\Theta)<br />\end{bmatrix}$