1) Consider any two body-fixed coordinate systems, F_(BI) and F_(B2) where F_(B2) is related to F_(01) by a positive rotation about the y_(11) axis through an angle (C) (a) Determine the nine angles between the axes of F_(a1) 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 _(x),Theta _(y) and Theta , Use this Information to write down T_(82,01) using the Euler angle representation of the direction cosine matrix. (c) Determine the eigenaxis and angle that relates F_(B2) to F_(B1) Use this Information to write down T_(B2,B1) using the Euler parameter representation of the direction cosine matrix.

(a) To determine the nine angles between the axes of $F_{a1}$ and $F_{B2}$, we need to consider the rotation that relates $F_{B2}$ to $F_{a1}$. Since $F_{B2}$ is related to $F_{a1}$ by a positive rotation about the $y_{11}$ axis through an angle (C), we can say that the rotation is of the form:<br /><br />$R = R_z(C)R_y(\alpha)R_x(\beta)$<br /><br />where $R_x$, $R_y$, and $R_z$ are the rotation matrices about the $x$, $y$, and $z$ axes, respectively, and $\alpha$ and $\beta$ are the angles between the $y$ and $z$ axes of $F_{a1}$ and $F_{B2}$, respectively.<br /><br />The nine angles between the axes of $F_{a1}$ and $F_{B2}$ are given by:<br /><br />$\alpha = \arccos(\langle y_{11}, y_{21}, y_{31} \rangle \cdot \langle y_{11}, y_{21}, y_{31} \rangle)$<br /><br />$\beta = \arccos(\langle z_{11}, z_{21}, z_{31} \rangle \cdot \langle z_{11}, z_{21}, z_{31} \rangle)$<br /><br />where $\langle \cdot \rangle$ denotes the dot product.<br /><br />Using this information, we can write down the direction cosine matrix $T_{B2,81}$ as:<br /><br />$T_{B2,81} = \begin{bmatrix}<br />\cos(\alpha) & 0 & -\sin(\alpha) \\<br />0 & 1 & 0 \\<br />\sin(\alpha) & 0 & \cos(\alpha)<br />\end{bmatrix}$<br /><br />(b) For a 321 Euler angle sequence from $F_{B1}$ to $F_{B2}$, we can determine the three angles $\Theta_x$, $\Theta_y$, and $\Theta_z$ as follows:<br /><br />$\Theta_x = \arccos(\langle x_{11}, x_{21}, x_{31} \rangle \cdot \langle x_{11}, x_{21}, x_{31} \rangle)$<br /><br />$\Theta_y = \arccos(\langle y_{11}, y_{21}, y_{31} \rangle \cdot \langle y_{11}, y_{21}, y_{31} \rangle)$<br /><br />$\Theta_z = \arccos(\langle z_{11}, z_{21}, z_{31} \rangle \cdot \langle z_{11}, z_{21}, z_{31} \rangle)$<br /><br />Using this information, we can write down the direction cosine matrix $T_{82,01}$ using the Euler angle representation as:<br /><br />$T_{82,01} = \begin{bmatrix}<br />\cos(\Theta_x) & -\sin(\Theta_x)\sin(\Theta_y) & \sin(\Theta_x)\sin(\Theta_y) \\<br />\sin(\Theta_x)\sin(\Theta_y) & \cos(\Theta_y) & -\sin(\Theta_x)\cos(\Theta_y) \\<br />-\sin(\Theta_x)\cos(\Theta_y) & \sin(\Theta_x)\cos(\Theta_y) & \cos(\Theta_x)\cos(\Theta_y)<br />\end{bmatrix}$<br /><br />(c) To determine the eigenaxis and angle that relates $F_{B2}$ to $F_{B1}$, we need to find the eigenvector and eigenvalue of the transformation matrix $T_{B2,B1}$. The eigenaxis is given by the eigenvector, and the angle is given by the argument of the eigenvalue.<br /><br />The transformation matrix $T_{B2,B1}$ can be written using the Euler parameter representation as:<br /><br />$T_{B2,B1} = e^{\begin{bmatrix}<br />0 & -\alpha & \beta \\<br />\alpha & 0 & -\gamma \\<br />-\\beta & \gamma & 0<br />\end{bmatrix}}$<br /><br />where $\alpha$, $\beta$, and $\gamma$ are the Euler parameters.<br /><br />Using this information, we can write down the direction cosine matrix $T_{B2,B1}$ using the Euler parameter representation as:<br /><br />$T_{B2,B1} = \begin{bmatrix}<br />\cos(\alpha) & -\sin(\alpha)\sin(\beta) & \sin(\alpha)\cos(\beta) \\<br />\sin(\alpha)\sin(\beta) & \cos(\beta) & -\sin(\alpha)\cos(\beta) \\<br />-\sin(\alpha)\cos(\beta) & \sin(\alpha)\cos(\beta) & \cos(\alpha)\cos(\beta)<br />\end{bmatrix}$