Problem 1 ( 30 points) 1- Find the components of the following vectors: $$ \begin{array}{1} \vec[u]=[1,0,0]^{T} W \vec{v}=[2,2,0]^{T} \ \vec[w]=[3,3,3][T) \endarray} $$ In a fixed frame $(x, y, z)$ after completing the series of positive rotations (counter clockwise) as follows: a. First rotate $90^{\circ]$ about the $y$-axis, then $99^{\circ]$ about the $*$-axis. b. First rotate $90^{\circ]$ about the $x$-axis, then $90^{\circ] $ about the $y$-axis c. First rotate $45^{\circ] $ about the $z$-axis, then $30^{\circ) $ about the $X$-axis 2- What is your comments about the results of rotations accomplished in $\mathbf {a}$ and $\mathbf {b}$ ? Note: In your solution, provide each rotation matrix that accomplishes these rotations in the given order for each case Problem 2 ( 30 points) Three coordinate systems $\left(\mathrm{x}_{0}, y_{0), \mathrm{z}_{0} ight), \left({\mathrm{x}_{1}, y_{1}, \mathrm{z}_{1} ight), \left\mathrm{x}_{2}, y_{2}, \mathrm{z}_{2} ight)$, with their origins at the same point, are related in the following way: - $\left(\mathrm{x}_{1}, y_{1}, z_{1} ight)$ is obtained from $\left(\mathrm{x}_{0), y(0), z_{0} ight)$ by a positive (counter clockwise) rotation of $90^{\circ $ about Sy_{0} -$ axis. - $\left\mathrm{x}_{2}, y_{2}, \mathrm{z}_{2} ight)$ is obtained from $\left\mathrm{x}_{1}, y_{1}, z_{1} ight)$ by a positive (counter clockwise) rotation of $90^{\circ]$ about $x_{1}$-axis. The same vectors components used in the previous problem $\vec{u}, \vec{v}, \ec{w}$ are expressed with respect to frame $\left(\mathrm{x}_{2}, y_{2}, \mathrm{z}_{2} ight)$. 1- Express each of these three vectors with respect to frame $\left(\mathrm{x}_{0}, y_{0}, \mathrm{z}_{@} ight)$. 2- Consider the previous problem and comment on your results. Note: In your solution, provide each rotation matrix that accomplishes these rotations in the given order for each case SP.SD. 334