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).