Exercise 3.2.13

In Example 3.46, we calculated the angle $\theta$ between a line and a plane by calculating the angle $\phi$ between the direction vector of the line and the normal vector of the plane according to the formula

and then taking $\theta=\frac{\pi}{2}-\phi$.

1. Explain what happens when the dot product is negative. How should we adjust the formula to ensure that $\theta$ is always between $0$ and $\frac{\pi}{2}$?

2. Show that one can get the answer in a single step with the formula

   $$\sin\theta=\frac{\left|\mathbf{n}\boldsymbol{\cdot}\mathbf{d}\right|}{\left\|\mathbf{n}\right\|\left\|\mathbf{d}\right\|}.$$