As shown in FIGURE 8.1.1(a), a spacecraft can perform rotations called pitch, roll, and yaw about three distinctaxes. To describe the coordinates of a point $P$ we use two coordinate systems: a fixed three-dimensional Cartesian coordinate system in which the coordinates of $P$ are $(x, y, z)$ and a spacecraft coordinate system that moves with the particular rotation. In Figure 8.1.1(b) we have illustrated a yaw - that is, a rotation around the $z$ -axis (which is perpendicular to the plane of the paper). The coordinates $\left(x_{Y}, y_{Y}, z_{Y}\right)$ of the point $P$ in the spacecraft system after the yaw are related to the coordinates $(x, y, z)$ of $P$ in the fixed coordinate system by the equations
$$
\begin{aligned}
&x_{Y}=x \cos \gamma+y \sin \gamma \\
&y_{Y}=-x \sin \gamma+y \cos \gamma \\
&z_{Y}=z
\end{aligned}
$$
where $\gamma$ is the angle of rotation.
(a) Verify that the foregoing system of equations can be written as the matrix equation
$$
\left(\begin{array}{l}
x_{Y} \\
y_{Y} \\
z_{Y}
\end{array}\right)=\mathbf{M}_{Y}\left(\begin{array}{l}
x \\
y \\
z
\end{array}\right)
$$