Calculus III Test: Chapter 13 Calculator Allowed Show all work on your own paper. Staple this sheet to the back. 1. If a  j  7k and b  2i  j  4k , find a  b . 2. Find an equation of the sphere with center 1, 4,3 and radius 5 . What is the intersection of this sphere with the xz plane? 3. Find a nonzero vector orthogonal to the plane through the points: P  0, 2,0 , Q  4,1, 2 , R  5,3,1

 1  4. Find a parametric equation and a symmetric equation for the line through points  0, ,1  and  2   2,1, 3 .

5. Find the point at which the line: x  3  t , y  2  t , z  5t intersects the plane: x  y  2 z  9

6. Sketch the graph of 25 x 2  4 y 2  z 2  100 as a surface in surface.

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. For an extra credit point, name the

7. If a  2,3, 6 and b  5, 1, 4 , find the scalar and vector projections of b onto a .

8. Find the volume of the parallelepiped determined by: a  6,3, 1 , b  0,1, 2 , c  4, 2,5

9. If a  i  2 j  2k and b  4i  3k , find the angle between the vectors to the nearest degree.

10. Find a unit vector that has the same direction as 4, 2, 4 .