Final answer:
The question does not provide enough specific information about direction angles to solve for the third direction angle of a vector. The concepts mentioned relate to vector angles and properties of direction cosines, but without specific values, no calculation can be made.
Step-by-step explanation:
The question provided does not directly give enough details about direction angles of a vector to find the third angle. Instead, it seems to be mixed with different vector addition problems and concepts including bearings and components of vectors. Generally, for a three-dimensional vector, the direction angles are associated with the axes (x, y, and z). If two direction angles are known, and if these angles are with respect to the x and y-axes, then one can use the relation that the sum of the squares of the cosines of the direction angles is equal to 1. This is because the direction cosines are related to the unit vector components of a vector. An obtuse or acute angle can be determined based on the calculations and understanding whether the vector lies in the respective quadrants, but specific values are not provided here to make calculations.
Regarding challenge problems and examples provided, they illustrate several principles of vector addition and vector angles, but do not directly answer the initial question. For instance, in the projectile range problem, a projectile's range is zero at both 90° (straight up) and 0° (horizontal to the ground, but without initial velocity, it goes nowhere). To find the angle between two vectors given their magnitudes and the magnitude of their sum or difference, one often uses the inverse cosine (arccos) of the dot product of the vectors divided by the product of their magnitudes, as stated. However, specific values for an angle or a conclusive answer to the original question are not provided.