Exercise 20

Consider the triangle in $\R^3$ formed by $P=(3, 2, -1)$, $Q=(1, -2, 4)$, and $R=(4, 4, 0)$.

1. Find $\overrightarrow{PQ}$ and $\overrightarrow{PR}$.

2. Observe that the area of $\triangle PQR$ is half of the area of the parallelogram formed by $\overrightarrow{PQ}$ and $\overrightarrow{PR}$. Hence find the area of $\triangle PQR$.

3. Find a unit vector that is orthogonal to the plane that contains points $P$, $Q$, and $R$.

4. Determine the measure of $\angle QPR$.