(Solved): D. Bryant Angles (123):xyz(123) - Textbook's Notation These angles are a ...

D. Bryant Angles $\left({?}_1?{?}_2?{?}_3\right):x?y?z(123)$ - Textbook's Notation These angles are also referred to as Cardan angles. The angular orientation of the body-fixed base ${\underline{\vec{e}}}^{\text{II }}$ is, again, represented as the result of a sequence of three rotations at the beginning of which the base ${\underline{\vec{e}}}^{\mathrm{II}}$ coincides with the reference base ${\underline{\vec{e}}}^{\mathrm{I}}$, (Fig. 2.3). $1^{\text{st }}$ rotation: Rotate axis ${\vec{e}}_1^{\mathrm{I}}$ with an angle ${?}_1$, result in the intermediate base ${\underline{\vec{e}}}^{II?}$. $2^{\text{nd }}$ rotation: Rotate axis ${\vec{e}}_2^{{\mathrm{II}}^{??}}$ with an angle ${?}_2$, produce intermediate base ${\underline{\vec{e}}}^{{\mathrm{II}}^{?}}$. $3^{\text{rd }}$ rotation: Rotate axis ${\vec{e}}_3^{{\mathrm{II}}^{?}}$ with an angle ${?}_3$, give the body-fixed base its final orientation denoted ${\underline{\underline{e}}}^{\mathrm{II}}$. The transformation equations for the individual rotations are Figure 2.3. Bryant angles ${?}_1,{?}_2,{?}_3$ ${\underline{\vec{e}}}^{\mathrm{II}}={\underline{A}}_3{\underline{\vec{e}}}^{{\mathrm{II}}^{?}}{\underline{\vec{e}}}^{{\mathrm{I}}^{?}}={\underline{A}}_2{\underline{\vec{e}}}^{{\mathrm{II}}^{??}}{\underline{\vec{e}}}^{\mathrm{I}{}^{??}}={\underline{A}}_1{\underline{\vec{e}}}^{\mathrm{I}}$ Problem 2.3. Determine all rotations $(\vec{n},?)$ which result in positions that can be produced by Bryant angles in the critical case $\cos {?}_2=0$.