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. [10 points] Use matrices and row operations to give a simple description for the set of solutions to the following system.

   $$\begin{array}{ccccccc}\hfill x_1& \hfill -& \hfill 2x_2& \hfill -& \hfill 9x_3& \hfill =& \hfill 5\\ \hfill 3x_1& \hfill +& \hfill x_2& \hfill -& \hfill 13x_3& \hfill =& \hfill 8\end{array}$$

2. Let $S$ be the sphere $x^2+3x+y^2-2y+z^2+4z=13$.

   1. [10 points] Find the center and radius of $S$.
   2. [6 points] Give a full geometric description of the curve obtained at the intersection of $S$ and the plane $z=0$.

    

3. [4 parts, 4 points each] Let $𝐚=⟨1,-5,2⟩$ and $𝐛=⟨1,-1,-2⟩$. Compute the following.

   1. $3𝐚-2𝐛$
       

   2. $|𝐚|$
       

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   3. $𝐚\cdot 𝐛$
       
   4. ${\mathrm{proj}}_{𝐚}𝐛$
       
4. [6 points] Find numbers $h$ and $k$ such that $⟨3,2,-5⟩$ and $⟨h,-1,k⟩$ are parallel.

5. Let $𝐚=⟨2,1,0⟩$ and $𝐛=⟨-1,3,5⟩$.

   1. [8 points] Find the angle between $𝐚$ and $𝐛$. Leave your answer in terms of inverse trigonometric functions.
   2. [2 points] Is the angle between $𝐚$ and $𝐛$ acute, obtuse, or right? Explain.

    

6. Let $T$ be the triangle with vertices $(2,1,0)$, $(3,1,4)$ and $(1,5,1)$.

   1. [10 points] Find a simple equation for the plane containing $T$.
   2. [6 points] Find the area of $T$.

7. [10 points] If the following lines intersect, then find the point of intersection. If the lines do not intersect, determine if they are parallel or skew.

   $$\begin{array}{llllllll}\hfill {𝐫}_1& =⟨9,3,-16⟩+t⟨1,1,4⟩& \hfill {𝐫}_2& =⟨5,14,-5⟩+t⟨-2,3,1⟩& \hfill & & \hfill & \end{array}$$

    

8. Consider the surface $\frac{x^2}{4}+y+\frac{z^2}{9}=1$.

   1. [6 points] Describe the axis-aligned traces.
   2. [10 points] Classify the surface and sketch it, labeling any significant points.