Each problem is worth 10 points, show all work and give adequate explanations for full credit. Please keep your work as legible as possible.

1. Write both vector and parametric equations for the line through (1,-4,3) and (3,0,2).
 
 

2. Find an equation for the plane containing the points (7,0,0), (0,3,0), and (0,0,10).
 
 

3. Despite extremely tight security, an orange is thrown from the origin with an initial velocity of 8i+19j+5k and subject to -9.8k acceleration. Give vector functions for the orange's velocity and position t seconds after being thrown.
 
 

4. Alvin the Ant walks in a spiral path up around an ice-cream cone so that his path is given by the vector function r(t)=<tcos t, tsin t, 3t>. Show that the length of the path Alvin traverses from t=0 to t=2 is given by .
 
 

5. Show that for any three-dimensional vectors a and b, ab is perpendicular to a.
 
 

6. For what value of x is the vector <-1,5,x> perpendicular to the vector <2,4,-3>?
 
 

7. Jonathan is a Calc student at Kansas State University, and he's having some trouble with cross products. Jonathan says "Man, I just can't handle this stuff. We had this test question about, like, when you do the cross product of two unit vectors does the answer have to be a unit vector. Well, I knew that if you do like i crossed with j then you get k, and that's a unit vector, so I said it had to work. When we got the test back I got, like, practically no credit, and it's driving me nuts so I can't even focus on the game tomorrow!"

Explain to Jonathan why his reasoning is or isn't valid (so he can go into the game with a clear head and get beat fairly).
 
 

8. Reduce the quadric x² + 8y = z² + 4y² +8 to standard form, classify the surface, and take a moment to visualize it mentally.
 
 

9. The helix r₁(t)=cos t i + sin t j + t k intersects the circle r₂(t)=cos t i + sin t j + 0 k at the point (1,0,0). Find the angle of intersection of these curves.
 
 

10. Find the curvature of the helix r(t)=(a cos t) i + (a sin t) j + bt k.
 
 

Extra Credit [5 points possible]:

Find the volume of the tetrahedron with vertices (0,0,0), (3,0,0), (0,1,1), and (2,-1,4). [It might help to recall our formula V = | a · (bc) | for the volume of a parallelepiped.]