MATH 260

EXAM 1

 

1.      (30 pt) Consider the points (1,2,-1), (2,2,0), (-3,1,-2).

a)      Compute the vectors a= and b=.

b)      Compute ab and ab.

c)      What is the angle between a and b?

d)      Compute the vector projection of a on b.

e)      Find the line determined by  and .

f)        Find the plane determined by all three points.

 

2.      (10 pt) Classify and sketch a picture of the quadratic surface .

3.      (20 pt) At time t=0 a particle is at the origin and has velocity vector . At any time, its acceleration is given by a.

a)      Find the vector-valued function that gives the position of the particle.

b)      How fast is the particle travelling when t=ln(2)?

 

4.      (10 pt) Find the point on the graph of   where the curvature is a maximum. (Justify that you have a maximum any way you want).

 

5.      (20 pt) Consider the set of all points in 3-space satisfying .

a)      Describe this object is 3-space.

b)      Rewrite the equation in terms of cylindrical and spherical coordinates.

 

6.      (10 pt) You travel along the vector valued function (is measured in miles). When you come to the point (1,2,1), you take off on the tangent vector and go for 5 miles. At what point in space do you end up?

 

7.      (10 pt)(EXTRA CREDIT). Show that the distance from the point  to the line  is given by the formula:

(Hint: Scalar projection).