Halliday, Resnick & Walker — Vectors: Problem 32

In Fig. 3.11, a cube of edge length \( a \) sits with one corner at the origin of an \( xyz \) coordinate system. A body diagonal is a line that extends from one corner to another through the center. In unit-vector notation, what is the body diagonal that extends from the corner at (a) coordinates \( (0, 0, 0) \), (b) coordinates \( (a, 0, 0) \), (c) coordinates \( (0, a, 0) \), and (d) coordinates \( (a, a, 0) \)? (e) Determine the angles that the body diagonals make with the adjacent edges. (f) Determine the length of the body diagonals in terms of \( a \).