Answer
See explanations.
Work Step by Step
Step 1. Identify the given quantities: $\alpha=20^{\circ}$ and $\beta=45^{\circ}$
Step 2. Recall the properties of the direction cosines $cos^2\alpha+cos^2\beta+cos^2\gamma=1$
Step 3. Test the given values for the above equation: $cos^220^{\circ}+cos^245^{\circ}+cos^2\gamma=1$ which gives $0.883+0.5+cos^2\gamma=1$ or $cos^2\gamma=-0.383$.
Step 4. Since $cos^2\gamma\geq0$. the above equation can not be true, which means that the given angles are impossible direction angles.
