Directions:

1. Section: Math251-007
2. Write your name with one character in each box below.
3. Show all work. No credit for answers without work.

1. [3 points] Let $P(2,-1,-2)$, $Q(3,0,3)$ and $R(1,2,-1)$ be points in ${ℝ}^3$. Find the angle formed by the triangle $\mathit{𝑃𝑄𝑅}$ at the vertex $Q$. Leave your answer in terms of inverse trig functions.

2. [2 parts, 3 points each] Let $𝐚=⟨1,-1,3⟩$ and $𝐛=⟨5,1,-2⟩$.

   1. Find $𝐚\times 𝐛$.
   2. Find the area of the triangle formed by placing the tails of the vectors $𝐚$ and $𝐛$ at the origin.
3. [1 point] Let $𝐚,𝐛\in {ℝ}^n$ and let $T$ be the triangle with vertices $0,𝐚,𝐛$. The cross product is defined only in $3$-dimensions, so it cannot be used to compute the area of $T$. Use the dot product theorem and the formula $\sin <!--\; FUNCTION\; APPLICATION\; -->𝜃=\sqrt{1-{\cos <!--\; FUNCTION\; APPLICATION\; -->}^2𝜃}$ for $0\le 𝜃\le \pi$ to find a formula for the area of $T$ in terms of the magnitudes of $𝐚$ and $𝐛$ and the dot product $𝐚\cdot 𝐛$.