Angle measurements and the trigonometric analysis of such measurements are used extensively in space science. Among the examples we shall consider here are some involving transformations between terrestrial (or celestial) and spacecraft coordinate systems, a variety of photogrammetric corrections, and the tracking of spacecraft from stations on Earth.

**PROBLEM 1.** A conventional right-handed three-dimensional spacecraft coordinate system is shown in Fig. 7.1. The angular motions of the spacecraft with respect to the x-, y-, and z-axes respectively are called roll, pitch, and yaw, shown in Fig. 7.1 by curved arrows. We shall develop the transformations between this coordinate system in a moving spacecraft and a reference coordinate system whose origin coincides with the one in the diagram but does not undergo rotation. Here, we shall consider a single rotation at a time. In Chapter 8, "Matrix Algebra," we shall investigate a series of such rotations.

When the spacecraft performs a rotation, the reference system remains fixed, but the spacecraft coordinate system undergoes the same rotation as the spacecraft. If the point Q has coordinates (x, y, z) in the reference system, we need to find its coordinates in the spacecraft system after such a rotation takes place. Let us consider each of the motions roll, pitch, and yaw separately.

**a. **Let the spacecraft coordinate system initially coincide with the reference system, and let the spacecraft undergo roll through angle R. Express the coordinates
(x sub r, y sub r, z sub r) of a point Q on the spacecraft in terms of (x, y, z) and R after this motion is performed.